Calculus Examples

$f (x) = x \sqrt{x - x^{4}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - x^{4}}$ as ${(x - x^{4})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(x - x^{4})}^{\frac{1}{2}}]$

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x - x^{4})}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(x - x^{4})}^{\frac{1}{2}}] + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - x^{4}$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x - x^{4}$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.3.3

Replace all occurrences of $u$ with $x - x^{4}$ .

$x (\frac{1}{2} {(x - x^{4})}^{\frac{1}{2} - 1} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x (\frac{1}{2} {(x - x^{4})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.5

Combine $- 1$ and $\frac{2}{2}$ .

$x (\frac{1}{2} {(x - x^{4})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.6

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(x - x^{4})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(x - x^{4})}^{\frac{1 - 2}{2}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.7.2

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(x - x^{4})}^{\frac{- 1}{2}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$x (\frac{1}{2} {(x - x^{4})}^{- \frac{1}{2}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8.2

Combine $\frac{1}{2}$ and ${(x - x^{4})}^{- \frac{1}{2}}$ .

$x (\frac{{(x - x^{4})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8.3

Move ${(x - x^{4})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(x - x^{4})}^{\frac{1}{2}}} \frac{d}{d x} [x - x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.8.4

Combine $\frac{1}{2 {(x - x^{4})}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \frac{d}{d x} [x - x^{4}] + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.9

By the Sum Rule, the derivative of $x - x^{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- x^{4}]$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (1 + \frac{d}{d x} [- x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (1 - \frac{d}{d x} [x^{4}]) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (1 - (4 x^{3})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.13

Multiply $4$ by $- 1$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (1 - 4 x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (1 - 4 x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot 1$

Step 1.15

Multiply ${(x - x^{4})}^{\frac{1}{2}}$ by $1$ .

$\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (1 - 4 x^{3}) + {(x - x^{4})}^{\frac{1}{2}}$

Step 1.16

Simplify.

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Step 1.16.1

Reorder terms.

$(1 - 4 x^{3}) \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} + {(x - x^{4})}^{\frac{1}{2}}$

Step 1.16.2

Multiply $1 - 4 x^{3}$ by $\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$\frac{(1 - 4 x^{3}) x}{2 {(x - x^{4})}^{\frac{1}{2}}} + {(x - x^{4})}^{\frac{1}{2}}$

Step 1.16.3

To write ${(x - x^{4})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$\frac{(1 - 4 x^{3}) x}{2 {(x - x^{4})}^{\frac{1}{2}}} + {(x - x^{4})}^{\frac{1}{2}} \cdot \frac{2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.4

Combine ${(x - x^{4})}^{\frac{1}{2}}$ and $\frac{2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$\frac{(1 - 4 x^{3}) x}{2 {(x - x^{4})}^{\frac{1}{2}}} + \frac{{(x - x^{4})}^{\frac{1}{2}} (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.5

Combine the numerators over the common denominator.

$\frac{(1 - 4 x^{3}) x + {(x - x^{4})}^{\frac{1}{2}} (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6

Simplify the numerator.

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Step 1.16.6.1

Apply the distributive property.

$\frac{1 x - 4 x^{3} x + {(x - x^{4})}^{\frac{1}{2}} \cdot 2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.2

Multiply $x$ by $1$ .

$\frac{x - 4 x^{3} x + {(x - x^{4})}^{\frac{1}{2}} \cdot 2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.16.6.3.1

Move $x$ .

$\frac{x - 4 (x \cdot x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot 2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.3.2

Multiply $x$ by $x^{3}$ .

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Step 1.16.6.3.2.1

Raise $x$ to the power of $1$ .

$\frac{x - 4 (x^{1} x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot 2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x - 4 x^{1 + 3} + {(x - x^{4})}^{\frac{1}{2}} \cdot 2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.3.3

Add $1$ and $3$ .

$\frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1}{2}} \cdot 2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.4

Multiply ${(x - x^{4})}^{\frac{1}{2}}$ by ${(x - x^{4})}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.16.6.4.1

Move ${(x - x^{4})}^{\frac{1}{2}}$ .

$\frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1}{2}} {(x - x^{4})}^{\frac{1}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.4.3

Combine the numerators over the common denominator.

$\frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1 + 1}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.4.4

Add $1$ and $1$ .

$\frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{2}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.4.5

Divide $2$ by $2$ .

$\frac{x - 4 x^{4} + {(x - x^{4})}^{1} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.5

Simplify ${(x - x^{4})}^{1} \cdot 2$ .

$\frac{x - 4 x^{4} + (x - x^{4}) \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.6

Apply the distributive property.

$\frac{x - 4 x^{4} + x \cdot 2 - x^{4} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.7

Move $2$ to the left of $x$ .

$\frac{x - 4 x^{4} + 2 \cdot x - x^{4} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.8

Multiply $2$ by $- 1$ .

$\frac{x - 4 x^{4} + 2 \cdot x - 2 x^{4}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.9

Add $x$ and $2 x$ .

$\frac{- 4 x^{4} + 3 x - 2 x^{4}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.10

Subtract $2 x^{4}$ from $- 4 x^{4}$ .

$\frac{- 6 x^{4} + 3 x}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.11

Factor $3 x$ out of $- 6 x^{4} + 3 x$ .

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Step 1.16.6.11.1

Factor $3 x$ out of $- 6 x^{4}$ .

$\frac{3 x (- 2 x^{3}) + 3 x}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.11.2

Factor $3 x$ out of $3 x$ .

$\frac{3 x (- 2 x^{3}) + 3 x (1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.6.11.3

Factor $3 x$ out of $3 x (- 2 x^{3}) + 3 x (1)$ .

$\frac{3 x (- 2 x^{3} + 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.7

Factor $- 1$ out of $- 2 x^{3}$ .

$\frac{3 x (- (2 x^{3}) + 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.8

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{3 x (- (2 x^{3}) - 1 (- 1))}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.9

Factor $- 1$ out of $- (2 x^{3}) - 1 (- 1)$ .

$\frac{3 x (- (2 x^{3} - 1))}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.10

Rewrite $- (2 x^{3} - 1)$ as $- 1 (2 x^{3} - 1)$ .

$\frac{3 x (- 1 (2 x^{3} - 1))}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.11

Move the negative in front of the fraction.

$- \frac{(3 x) (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 1.16.12

Reorder factors in $- \frac{(3 x) (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [\frac{x (2 x^{3} - 1)}{{(x - x^{4})}^{\frac{1}{2}}}]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{d}{d x} \frac{x (2 x^{3} - 1)}{{(x - x^{4})}^{\frac{1}{2}}}$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x (2 x^{3} - 1)$ and $g (x) = {(x - x^{4})}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{3} - 1)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{{({(x - x^{4})}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(x - x^{4})}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{3} - 1)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{{(x - x^{4})}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{3} - 1)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{{(x - x^{4})}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{3} - 1)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.4

Simplify.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{3} - 1)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 2 x^{3} - 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x \frac{d}{d x} (2 x^{3} - 1) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.6

Differentiate.

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Step 2.6.1

By the Sum Rule, the derivative of $2 x^{3} - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 1]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x (\frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (- 1)) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.6.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x (2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 1)) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.6.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x (2 (3 x^{2}) + \frac{d}{d x} (- 1)) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.6.4

Multiply $3$ by $2$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x (6 x^{2} + \frac{d}{d x} (- 1)) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.6.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x (6 x^{2} + 0) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.6.6

Add $6 x^{2}$ and $0$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (x (6 x^{2}) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.7

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (6 (x \cdot x^{2}) + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (6 x^{1 + 2} + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.9

Differentiate using the Power Rule.

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Step 2.9.1

Add $1$ and $2$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (6 x^{3} + (2 x^{3} - 1) \frac{d}{d x} (x)) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.9.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (6 x^{3} + (2 x^{3} - 1) \cdot 1) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.9.3

Simplify by adding terms.

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Step 2.9.3.1

Multiply $2 x^{3} - 1$ by $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (6 x^{3} + 2 x^{3} - 1) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.9.3.2

Add $6 x^{3}$ and $2 x^{3}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) \frac{d}{d x} {(x - x^{4})}^{\frac{1}{2}}}{x - x^{4}}$

Step 2.10

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - x^{4}$ .

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Step 2.10.1

To apply the Chain Rule, set $u_{1}$ as $x - x^{4}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.10.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.10.3

Replace all occurrences of $u_{1}$ with $x - x^{4}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1}{2} - 1} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.12

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.13

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.14

Simplify the numerator.

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Step 2.14.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1 - 2}{2}} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.14.2

Subtract $2$ from $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{- 1}{2}} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.15

Combine fractions.

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Step 2.15.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2} \cdot {(x - x^{4})}^{- \frac{1}{2}} \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.15.2

Combine $\frac{1}{2}$ and ${(x - x^{4})}^{- \frac{1}{2}}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{{(x - x^{4})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.15.3

Move ${(x - x^{4})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - x (2 x^{3} - 1) (\frac{1}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x - x^{4}))}{x - x^{4}}$

Step 2.15.4

Combine $\frac{1}{2 {(x - x^{4})}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (2 x^{3} - 1) \frac{d}{d x} (x - x^{4})}{x - x^{4}}$

Step 2.16

By the Sum Rule, the derivative of $x - x^{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- x^{4}]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (\frac{d}{d x} (x) + \frac{d}{d x} (- x^{4})))}{x - x^{4}}$

Step 2.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 + \frac{d}{d x} (- x^{4})))}{x - x^{4}}$

Step 2.18

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - \frac{d}{d x} x^{4}))}{x - x^{4}}$

Step 2.19

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - (4 x^{3})))}{x - x^{4}}$

Step 2.20

Combine fractions.

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Step 2.20.1

Multiply $4$ by $- 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3}))}{x - x^{4}}$

Step 2.20.2

Multiply $\frac{{(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (2 x^{3} - 1) (1 - 4 x^{3})}{x - x^{4}}$ by $\frac{3}{2}$ .

$f'' (x) = - \frac{({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3}))) \cdot 3}{(x - x^{4}) \cdot 2}$

Step 2.20.3

Reorder.

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Step 2.20.3.1

Move $3$ to the left of ${(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (2 x^{3} - 1) (1 - 4 x^{3})$ .

$f'' (x) = - \frac{3 \cdot ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{(x - x^{4}) \cdot 2}$

Step 2.20.3.2

Move $2$ to the left of $x - x^{4}$ .

$f'' (x) = - \frac{3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 \cdot (x - x^{4})}$

Step 2.21

Simplify.

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Step 2.21.1

Apply the distributive property.

$f'' (x) = - \frac{3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1)) + 3 (- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 (x - x^{4})}$

Step 2.21.2

Apply the distributive property.

$f'' (x) = - \frac{3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1)) + 3 (- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3

Simplify the numerator.

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Step 2.21.3.1

Factor $3$ out of $3 {(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) + 3 (- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (2 x^{3} - 1) (1 - 4 x^{3}))$ .

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Step 2.21.3.1.1

Factor $3$ out of $3 {(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1)$ .

$f'' (x) = - \frac{3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1)) + 3 (- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3.1.2

Factor $3$ out of $3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1)) + 3 (- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} (2 x^{3} - 1) (1 - 4 x^{3}))$ .

$f'' (x) = - \frac{3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3} - 1) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3.2

Apply the distributive property.

$f'' (x) = - \frac{3 ({(x - x^{4})}^{\frac{1}{2}} (8 x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot - 1 - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} + {(x - x^{4})}^{\frac{1}{2}} \cdot - 1 - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3.4

Move $- 1$ to the left of ${(x - x^{4})}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - 1 \cdot {(x - x^{4})}^{\frac{1}{2}} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3.5

Rewrite $- 1 {(x - x^{4})}^{\frac{1}{2}}$ as $- {(x - x^{4})}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot ((2 x^{3} - 1) (1 - 4 x^{3})))}{2 x + 2 (- x^{4})}$

Step 2.21.3.6

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (2 x^{3}) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.7

Cancel the common factor of $2$ .

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Step 2.21.3.7.1

Move the leading negative in $- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}$ into the numerator.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (2 x^{3}) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.7.2

Factor $2$ out of $2 {(x - x^{4})}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x}{2 ({(x - x^{4})}^{\frac{1}{2}})} \cdot (2 x^{3}) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.7.3

Factor $2$ out of $2 x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x}{2 ({(x - x^{4})}^{\frac{1}{2}})} \cdot (2 (x^{3})) - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.7.4

Cancel the common factor.

Step 2.21.3.7.5

Rewrite the expression.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x}{{(x - x^{4})}^{\frac{1}{2}}} \cdot x^{3} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.8

Combine $\frac{- x}{{(x - x^{4})}^{\frac{1}{2}}}$ and $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x \cdot x^{3}}{{(x - x^{4})}^{\frac{1}{2}}} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.9

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.9.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- (x^{3} x)}{{(x - x^{4})}^{\frac{1}{2}}} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.9.2

Multiply $x^{3}$ by $x$ .

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Step 2.21.3.9.2.1

Raise $x$ to the power of $1$ .

Step 2.21.3.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x^{3 + 1}}{{(x - x^{4})}^{\frac{1}{2}}} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.9.3

Add $3$ and $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} - \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.10

Multiply $- \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot - 1$ .

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Step 2.21.3.10.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + 1 (\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}})) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.10.2

Multiply $\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}$ by $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (\frac{- x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + (- \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}) (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.12

Expand $(- \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}) (1 - 4 x^{3})$ using the FOIL Method.

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Step 2.21.3.12.1

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - 4 x^{3}) + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.12.2

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} \cdot 1 - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}) + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.12.3

Apply the distributive property.

Step 2.21.3.13

Simplify and combine like terms.

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Step 2.21.3.13.1

Simplify each term.

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Step 2.21.3.13.1.1

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}) + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.2

Multiply $- \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} (- 4 x^{3})$ .

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Step 2.21.3.13.1.2.1

Multiply $- 4$ by $- 1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + 4 (\frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}}) \cdot x^{3} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.2.2

Combine $4$ and $\frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} \cdot x^{3} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.2.3

Combine $\frac{4 x^{4}}{{(x - x^{4})}^{\frac{1}{2}}}$ and $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{4} x^{3}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.2.4

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.13.1.2.4.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 (x^{3} x^{4})}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.2.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{3 + 4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.2.4.3

Add $3$ and $4$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot 1 + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.3

Multiply $\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}$ by $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (- 4 x^{3}))}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - 4 \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot x^{3})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.5

Cancel the common factor of $2$ .

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Step 2.21.3.13.1.5.1

Factor $2$ out of $- 4$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} + 2 \cdot (- 2 \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}) \cdot x^{3})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.5.2

Cancel the common factor.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} + 2 \cdot (- 2 \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}) \cdot x^{3})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.5.3

Rewrite the expression.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - 2 \frac{x}{{(x - x^{4})}^{\frac{1}{2}}} \cdot x^{3})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.6

Combine $- 2$ and $\frac{x}{{(x - x^{4})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} + \frac{- 2 x}{{(x - x^{4})}^{\frac{1}{2}}} \cdot x^{3})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.7

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{(2) x}{{(x - x^{4})}^{\frac{1}{2}}} \cdot x^{3})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.8

Multiply $- \frac{2 x}{{(x - x^{4})}^{\frac{1}{2}}} x^{3}$ .

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Step 2.21.3.13.1.8.1

Combine $x^{3}$ and $\frac{2 x}{{(x - x^{4})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{x^{3} (2 x)}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.8.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.21.3.13.1.8.2.1

Move $x$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{x \cdot x^{3} \cdot 2}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.8.2.2

Multiply $x$ by $x^{3}$ .

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Step 2.21.3.13.1.8.2.2.1

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{x \cdot x^{3} \cdot 2}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.8.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{x^{1 + 3} \cdot 2}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.8.2.3

Add $1$ and $3$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{x^{4} \cdot 2}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.1.9

Move $2$ to the left of $x^{4}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} - \frac{x^{4}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} - \frac{2 x^{4}}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.13.2

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{4 x^{7}}{{(x - x^{4})}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} + \frac{- x^{4} - 2 x^{4}}{{(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.14

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{4 x^{7} - x^{4} - 2 x^{4}}{{(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.15

Subtract $2 x^{4}$ from $- x^{4}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{4 x^{7} - 3 x^{4}}{{(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.16

Factor $x^{4}$ out of $4 x^{7} - 3 x^{4}$ .

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Step 2.21.3.16.1

Factor $x^{4}$ out of $4 x^{7}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3}) - 3 x^{4}}{{(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.16.2

Factor $x^{4}$ out of $- 3 x^{4}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3}) + x^{4} \cdot - 3}{{(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.16.3

Factor $x^{4}$ out of $x^{4} (4 x^{3}) + x^{4} \cdot - 3$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3} - 3)}{{(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.17

Reorder terms.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3} - 3)}{{(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.18

To write $\frac{x^{4} (4 x^{3} - 3)}{{(- x^{4} + x)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3} - 3)}{{(- x^{4} + x)}^{\frac{1}{2}}} \cdot \frac{2}{2} + \frac{x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.19

Write each expression with a common denominator of $2 {(- x^{4} + x)}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.21.3.19.1

Multiply $\frac{x^{4} (4 x^{3} - 3)}{{(- x^{4} + x)}^{\frac{1}{2}}}$ by $\frac{2}{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3} - 3) \cdot 2}{{(- x^{4} + x)}^{\frac{1}{2}} \cdot 2} + \frac{x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.19.2

Move $2$ to the left of ${(- x^{4} + x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3} - 3) \cdot 2}{2 {(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.20

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x^{4} (4 x^{3} - 3) \cdot 2 + x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21

Simplify the numerator.

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Step 2.21.3.21.1

Factor $x$ out of $x^{4} (4 x^{3} - 3) \cdot 2 + x$ .

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Step 2.21.3.21.1.1

Factor $x$ out of $x^{4} (4 x^{3} - 3) \cdot 2$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((x^{3} (4 x^{3} - 3)) \cdot 2) + x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.1.2

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((x^{3} (4 x^{3} - 3)) \cdot 2) + x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.1.3

Factor $x$ out of $x^{1}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((x^{3} (4 x^{3} - 3)) \cdot 2) + x \cdot 1}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.1.4

Factor $x$ out of $x ((x^{3} (4 x^{3} - 3)) \cdot 2) + x \cdot 1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((x^{3} (4 x^{3} - 3)) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.2

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((x^{3} (4 x^{3}) + x^{3} \cdot - 3) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 x^{3} x^{3} + x^{3} \cdot - 3) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.4

Move $- 3$ to the left of $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 x^{3} x^{3} - 3 \cdot x^{3}) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.5

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.21.5.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 (x^{3} x^{3}) - 3 \cdot x^{3}) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 x^{3 + 3} - 3 \cdot x^{3}) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.5.3

Add $3$ and $3$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 x^{6} - 3 \cdot x^{3}) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 x^{6} - 3 x^{3}) \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.6

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (4 x^{6} \cdot 2 - 3 x^{3} \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.7

Multiply $2$ by $4$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 x^{6} - 3 x^{3} \cdot 2 + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.8

Multiply $2$ by $- 3$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 x^{6} - 6 x^{3} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9

Rewrite $8 x^{6} - 6 x^{3} + 1$ in a factored form.

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Step 2.21.3.21.9.1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 {(x^{3})}^{2} - 6 x^{3} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.2

Let $u_{2} = x^{3}$ . Substitute $u_{2}$ for all occurrences of $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 {u_{2}}^{2} - 6 u_{2} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.3

Factor by grouping.

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Step 2.21.3.21.9.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot 1 = 8$ and whose sum is $b = - 6$ .

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Step 2.21.3.21.9.3.1.1

Factor $- 6$ out of $- 6 u_{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 {u_{2}}^{2} - 6 u_{2} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.3.1.2

Rewrite $- 6$ as $- 2$ plus $- 4$

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 {u_{2}}^{2} + (- 2 - 4) u_{2} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.3.1.3

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (8 {u_{2}}^{2} - 2 u_{2} - 4 u_{2} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.3.2

Factor out the greatest common factor from each group.

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Step 2.21.3.21.9.3.2.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((8 {u_{2}}^{2} - 2 u_{2}) - 4 u_{2} + 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.3.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (2 u_{2} (4 u_{2} - 1) - (4 u_{2} - 1))}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.3.3

Factor the polynomial by factoring out the greatest common factor, $4 u_{2} - 1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 u_{2} - 1) (2 u_{2} - 1))}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.21.9.4

Replace all occurrences of $u_{2}$ with $x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x ((4 x^{3} - 1) (2 x^{3} - 1))}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} - {(x - x^{4})}^{\frac{1}{2}} + \frac{x (4 x^{3} - 1) (2 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.22

To write $8 {(x - x^{4})}^{\frac{1}{2}} x^{3}$ as a fraction with a common denominator, multiply by $\frac{2 {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{4})}^{\frac{1}{2}} x^{3} \cdot \frac{2 {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x (4 x^{3} - 1) (2 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}} - {(x - x^{4})}^{\frac{1}{2}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.23

Combine $8 {(x - x^{4})}^{\frac{1}{2}} x^{3}$ and $\frac{2 {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{4})}^{\frac{1}{2}} x^{3} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}} + \frac{x (4 x^{3} - 1) (2 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}} - {(x - x^{4})}^{\frac{1}{2}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.24

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{4})}^{\frac{1}{2}} x^{3} (2 {(- x^{4} + x)}^{\frac{1}{2}}) + x (4 x^{3} - 1) (2 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}} - {(x - x^{4})}^{\frac{1}{2}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.25

To write $- {(x - x^{4})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ .

Step 2.21.3.26

Combine $- {(x - x^{4})}^{\frac{1}{2}}$ and $\frac{2 {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{4})}^{\frac{1}{2}} x^{3} (2 {(- x^{4} + x)}^{\frac{1}{2}}) + x (4 x^{3} - 1) (2 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}} + \frac{- {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.27

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{4})}^{\frac{1}{2}} x^{3} (2 {(- x^{4} + x)}^{\frac{1}{2}}) + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28

Rewrite $\frac{8 {(x - x^{4})}^{\frac{1}{2}} x^{3} (2 {(- x^{4} + x)}^{\frac{1}{2}}) + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.21.3.28.1

Multiply $8 {(x - x^{4})}^{\frac{1}{2}} x^{3} (2 {(- x^{4} + x)}^{\frac{1}{2}})$ .

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Step 2.21.3.28.1.1

Multiply $2$ by $8$ .

$f'' (x) = - \frac{3 (\frac{16 {(x - x^{4})}^{\frac{1}{2}} x^{3} {(- x^{4} + x)}^{\frac{1}{2}} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.2

Reorder terms.

$f'' (x) = - \frac{3 (\frac{16 {(- x^{4} + x)}^{\frac{1}{2}} x^{3} {(- x^{4} + x)}^{\frac{1}{2}} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.3

Multiply ${(- x^{4} + x)}^{\frac{1}{2}}$ by ${(- x^{4} + x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.21.3.28.1.3.1

Move ${(- x^{4} + x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (\frac{16 ({(- x^{4} + x)}^{\frac{1}{2}} {(- x^{4} + x)}^{\frac{1}{2}}) x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{16 {(- x^{4} + x)}^{\frac{1}{2} + \frac{1}{2}} x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.3.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{16 {(- x^{4} + x)}^{\frac{1 + 1}{2}} x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.3.4

Add $1$ and $1$ .

$f'' (x) = - \frac{3 (\frac{16 {(- x^{4} + x)}^{\frac{2}{2}} x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.3.5

Divide $2$ by $2$ .

$f'' (x) = - \frac{3 (\frac{16 (- x^{4} + x) x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.1.4

Simplify $16 {(- x^{4} + x)}^{1} x^{3}$ .

Step 2.21.3.28.2

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{(16 (- x^{4}) + 16 x) x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.3

Multiply $- 1$ by $16$ .

$f'' (x) = - \frac{3 (\frac{(- 16 x^{4} + 16 x) x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.4

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{4} x^{3} + 16 x \cdot x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.5

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.28.5.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (\frac{- 16 (x^{3} x^{4}) + 16 x \cdot x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3 + 4} + 16 x \cdot x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.5.3

Add $3$ and $4$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x \cdot x^{3} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.6

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.28.6.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 (x^{3} x) + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.6.2

Multiply $x^{3}$ by $x$ .

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Step 2.21.3.28.6.2.1

Raise $x$ to the power of $1$ .

Step 2.21.3.28.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{3 + 1} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.6.3

Add $3$ and $1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + x (4 x^{3} - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.7

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (x (4 x^{3}) + x \cdot - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.8

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (4 x \cdot x^{3} + x \cdot - 1) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.9

Move $- 1$ to the left of $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (4 x \cdot x^{3} - 1 \cdot x) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.10

Simplify each term.

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Step 2.21.3.28.10.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.28.10.1.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (4 (x^{3} x) - 1 \cdot x) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.10.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.21.3.28.10.1.2.1

Raise $x$ to the power of $1$ .

Step 2.21.3.28.10.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (4 x^{3 + 1} - 1 \cdot x) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.10.1.3

Add $3$ and $1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (4 x^{4} - 1 \cdot x) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.10.2

Rewrite $- 1 x$ as $- x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + (4 x^{4} - x) (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.11

Expand $(4 x^{4} - x) (2 x^{3} - 1)$ using the FOIL Method.

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Step 2.21.3.28.11.1

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 x^{4} (2 x^{3} - 1) - x (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.11.2

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 x^{4} (2 x^{3}) + 4 x^{4} \cdot - 1 - x (2 x^{3} - 1) - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.11.3

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 x^{4} (2 x^{3}) + 4 x^{4} \cdot - 1 - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12

Simplify and combine like terms.

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Step 2.21.3.28.12.1

Simplify each term.

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Step 2.21.3.28.12.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 \cdot (2 x^{4} x^{3}) + 4 x^{4} \cdot - 1 - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.28.12.1.2.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 \cdot (2 (x^{3} x^{4})) + 4 x^{4} \cdot - 1 - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 \cdot (2 x^{3 + 4}) + 4 x^{4} \cdot - 1 - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.2.3

Add $3$ and $4$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 4 \cdot (2 x^{7}) + 4 x^{4} \cdot - 1 - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.3

Multiply $4$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} + 4 x^{4} \cdot - 1 - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.4

Multiply $- 1$ by $4$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - x (2 x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 1 \cdot (2 x \cdot x^{3}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.6

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.21.3.28.12.1.6.1

Move $x^{3}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 1 \cdot (2 (x^{3} x)) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.6.2

Multiply $x^{3}$ by $x$ .

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Step 2.21.3.28.12.1.6.2.1

Raise $x$ to the power of $1$ .

Step 2.21.3.28.12.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 1 \cdot (2 x^{3 + 1}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.6.3

Add $3$ and $1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 1 \cdot (2 x^{4}) - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.7

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 2 x^{4} - x \cdot - 1 - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.8

Multiply $- x \cdot - 1$ .

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Step 2.21.3.28.12.1.8.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 2 x^{4} + 1 x - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.1.8.2

Multiply $x$ by $1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 4 x^{4} - 2 x^{4} + x - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.12.2

Subtract $2 x^{4}$ from $- 4 x^{4}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - {(x - x^{4})}^{\frac{1}{2}} (2 {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.13

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 1 \cdot (2 {(x - x^{4})}^{\frac{1}{2}} {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.14

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 {(x - x^{4})}^{\frac{1}{2}} {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15

Multiply $- 2 {(x - x^{4})}^{\frac{1}{2}} {(- x^{4} + x)}^{\frac{1}{2}}$ .

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Step 2.21.3.28.15.1

Reorder terms.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 {(- x^{4} + x)}^{\frac{1}{2}} {(- x^{4} + x)}^{\frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15.2

Multiply ${(- x^{4} + x)}^{\frac{1}{2}}$ by ${(- x^{4} + x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.21.3.28.15.2.1

Move ${(- x^{4} + x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 ({(- x^{4} + x)}^{\frac{1}{2}} {(- x^{4} + x)}^{\frac{1}{2}})}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 {(- x^{4} + x)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15.2.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 {(- x^{4} + x)}^{\frac{1 + 1}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15.2.4

Add $1$ and $1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 {(- x^{4} + x)}^{\frac{2}{2}}}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15.2.5

Divide $2$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 (- x^{4} + x)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.15.3

Simplify $- 2 {(- x^{4} + x)}^{1}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 (- x^{4} + x)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.16

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x - 2 (- x^{4}) - 2 x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.17

Multiply $- 1$ by $- 2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{7} + 16 x^{4} + 8 x^{7} - 6 x^{4} + x + 2 x^{4} - 2 x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.18

Add $- 16 x^{7}$ and $8 x^{7}$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{7} + 16 x^{4} - 6 x^{4} + x + 2 x^{4} - 2 x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.19

Subtract $6 x^{4}$ from $16 x^{4}$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{7} + 10 x^{4} + x + 2 x^{4} - 2 x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.20

Add $10 x^{4}$ and $2 x^{4}$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{7} + 12 x^{4} + x - 2 x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.21

Subtract $2 x$ from $x$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{7} + 12 x^{4} - x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.22

Factor $x$ out of $- 8 x^{7} + 12 x^{4} - x$ .

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Step 2.21.3.28.22.1

Factor $x$ out of $- 8 x^{7}$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{6}) + 12 x^{4} - x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.22.2

Factor $x$ out of $12 x^{4}$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{6}) + x (12 x^{3}) - x}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.22.3

Factor $x$ out of $- x$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{6}) + x (12 x^{3}) + x \cdot - 1}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.22.4

Factor $x$ out of $x (- 8 x^{6}) + x (12 x^{3})$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{6} + 12 x^{3}) + x \cdot - 1}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.3.28.22.5

Factor $x$ out of $x (- 8 x^{6} + 12 x^{3}) + x \cdot - 1$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{4})}$

Step 2.21.4

Combine terms.

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Step 2.21.4.1

Combine $3$ and $\frac{x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{\frac{3 (x (- 8 x^{6} + 12 x^{3} - 1))}{2 {(- x^{4} + x)}^{\frac{1}{2}}}}{2 x + 2 (- x^{4})}$

Step 2.21.4.2

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{\frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}}}{2 x - 2 x^{4}}$

Step 2.21.4.3

Rewrite $\frac{\frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}}}{2 x - 2 x^{4}}$ as a product.

$f'' (x) = - (\frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}} \cdot \frac{1}{2 x - 2 x^{4}})$

Step 2.21.4.4

Multiply $\frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}}}$ by $\frac{1}{2 x - 2 x^{4}}$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} (2 x - 2 x^{4})}$

Step 2.21.5

Simplify the denominator.

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Step 2.21.5.1

Factor $2 x$ out of $2 x - 2 x^{4}$ .

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Step 2.21.5.1.1

Factor $2 x$ out of $2 x$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} (2 x (1) - 2 x^{4})}$

Step 2.21.5.1.2

Factor $2 x$ out of $- 2 x^{4}$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} (2 x (1) + 2 x (- x^{3}))}$

Step 2.21.5.1.3

Factor $2 x$ out of $2 x (1) + 2 x (- x^{3})$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} (2 x (1 - x^{3}))}$

Step 2.21.5.2

Rewrite $1$ as $1^{3}$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} (2 x (1^{3} - x^{3}))}$

Step 2.21.5.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} (2 x ((1 - x) (1^{2} + 1 x + x^{2})))}$

Step 2.21.5.4

Factor.

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{2 {(- x^{4} + x)}^{\frac{1}{2}} \cdot (2 x (1 - x) (1 + x + x^{2}))}$

Step 2.21.5.5

Multiply $2$ by $2$ .

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} x (1 - x) (1 + x + x^{2})}$

Step 2.21.6

Cancel the common factor.

$f'' (x) = - \frac{3 x (- 8 x^{6} + 12 x^{3} - 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} x (1 - x) (1 + x + x^{2})}$

Step 2.21.7

Rewrite the expression.

$f'' (x) = - \frac{3 (- 8 x^{6} + 12 x^{3} - 1)}{(4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x)) (1 + x + x^{2})}$

Step 2.21.8

Factor $- 1$ out of $- 8 x^{6}$ .

$f'' (x) = - \frac{3 (- (8 x^{6}) + 12 x^{3} - 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.9

Factor $- 1$ out of $12 x^{3}$ .

$f'' (x) = - \frac{3 (- (8 x^{6}) - (- 12 x^{3}) - 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.10

Factor $- 1$ out of $- (8 x^{6}) - (- 12 x^{3})$ .

$f'' (x) = - \frac{3 (- (8 x^{6} - 12 x^{3}) - 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.11

Rewrite $- 1$ as $- 1 (1)$ .

$f'' (x) = - \frac{3 (- (8 x^{6} - 12 x^{3}) - 1 \cdot 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.12

Factor $- 1$ out of $- (8 x^{6} - 12 x^{3}) - 1 (1)$ .

$f'' (x) = - \frac{3 (- (8 x^{6} - 12 x^{3} + 1))}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.13

Rewrite $- (8 x^{6} - 12 x^{3} + 1)$ as $- 1 (8 x^{6} - 12 x^{3} + 1)$ .

$f'' (x) = - \frac{3 (- 1 (8 x^{6} - 12 x^{3} + 1))}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.14

Move the negative in front of the fraction.

$f'' (x) = \frac{3 (8 x^{6} - 12 x^{3} + 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 2.21.15

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{3 (8 x^{6} - 12 x^{3} + 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})})$

Step 2.21.16

Multiply $\frac{3 (8 x^{6} - 12 x^{3} + 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$ by $1$ .

$f'' (x) = \frac{3 (8 x^{6} - 12 x^{3} + 1)}{4 {(- x^{4} + x)}^{\frac{1}{2}} (1 - x) (1 + x + x^{2})}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - x^{4}}$ as ${(x - x^{4})}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x {(x - x^{4})}^{\frac{1}{2}})$

Step 4.1.2

$f' (x) = x \frac{d}{d x} ({(x - x^{4})}^{\frac{1}{2}}) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - x^{4}$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $x - x^{4}$ .

$f' (x) = x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$f' (x) = x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.3.3

Replace all occurrences of $u$ with $x - x^{4}$ .

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1}{2} - 1} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.5

Combine $- 1$ and $\frac{2}{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.6

Combine the numerators over the common denominator.

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{1 - 2}{2}} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.7.2

Subtract $2$ from $1$ .

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{\frac{- 1}{2}} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

$f' (x) = x (\frac{1}{2} \cdot {(x - x^{4})}^{- \frac{1}{2}} \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.8.2

Combine $\frac{1}{2}$ and ${(x - x^{4})}^{- \frac{1}{2}}$ .

$f' (x) = x (\frac{{(x - x^{4})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.8.3

Move ${(x - x^{4})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x (\frac{1}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x - x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.8.4

Combine $\frac{1}{2 {(x - x^{4})}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x - x^{4}) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.9

By the Sum Rule, the derivative of $x - x^{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- x^{4}]$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (- x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 + \frac{d}{d x} (- x^{4})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - \frac{d}{d x} x^{4}) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - (4 x^{3})) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.13

Multiply $4$ by $- 1$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - 4 x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 4.1.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - 4 x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot 1$

Step 4.1.15

Multiply ${(x - x^{4})}^{\frac{1}{2}}$ by $1$ .

$f' (x) = \frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}} \cdot (1 - 4 x^{3}) + {(x - x^{4})}^{\frac{1}{2}}$

Step 4.1.16

Simplify.

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Step 4.1.16.1

Reorder terms.

$f' (x) = (1 - 4 x^{3}) (\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}) + {(x - x^{4})}^{\frac{1}{2}}$

Step 4.1.16.2

Multiply $1 - 4 x^{3}$ by $\frac{x}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$f' (x) = \frac{(1 - 4 x^{3}) x}{2 {(x - x^{4})}^{\frac{1}{2}}} + {(x - x^{4})}^{\frac{1}{2}}$

Step 4.1.16.3

To write ${(x - x^{4})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$f' (x) = \frac{(1 - 4 x^{3}) x}{2 {(x - x^{4})}^{\frac{1}{2}}} + {(x - x^{4})}^{\frac{1}{2}} \cdot \frac{2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.4

Combine ${(x - x^{4})}^{\frac{1}{2}}$ and $\frac{2 {(x - x^{4})}^{\frac{1}{2}}}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$f' (x) = \frac{(1 - 4 x^{3}) x}{2 {(x - x^{4})}^{\frac{1}{2}}} + \frac{{(x - x^{4})}^{\frac{1}{2}} (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.5

Combine the numerators over the common denominator.

$f' (x) = \frac{(1 - 4 x^{3}) x + {(x - x^{4})}^{\frac{1}{2}} (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6

Simplify the numerator.

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Step 4.1.16.6.1

Apply the distributive property.

$f' (x) = \frac{1 x - 4 x^{3} x + {(x - x^{4})}^{\frac{1}{2}} \cdot (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.2

Multiply $x$ by $1$ .

$f' (x) = \frac{x - 4 x^{3} x + {(x - x^{4})}^{\frac{1}{2}} \cdot (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.1.16.6.3.1

Move $x$ .

$f' (x) = \frac{x - 4 (x \cdot x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.3.2

Multiply $x$ by $x^{3}$ .

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Step 4.1.16.6.3.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{x - 4 (x \cdot x^{3}) + {(x - x^{4})}^{\frac{1}{2}} \cdot (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{x - 4 x^{1 + 3} + {(x - x^{4})}^{\frac{1}{2}} \cdot (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.3.3

Add $1$ and $3$ .

$f' (x) = \frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1}{2}} \cdot (2 {(x - x^{4})}^{\frac{1}{2}})}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.4

Multiply ${(x - x^{4})}^{\frac{1}{2}}$ by ${(x - x^{4})}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1.16.6.4.1

Move ${(x - x^{4})}^{\frac{1}{2}}$ .

$f' (x) = \frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1}{2}} {(x - x^{4})}^{\frac{1}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.4.3

Combine the numerators over the common denominator.

$f' (x) = \frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{1 + 1}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.4.4

Add $1$ and $1$ .

$f' (x) = \frac{x - 4 x^{4} + {(x - x^{4})}^{\frac{2}{2}} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.4.5

Divide $2$ by $2$ .

$f' (x) = \frac{x - 4 x^{4} + (x - x^{4}) \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.5

Simplify ${(x - x^{4})}^{1} \cdot 2$ .

$f' (x) = \frac{x - 4 x^{4} + (x - x^{4}) \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.6

Apply the distributive property.

$f' (x) = \frac{x - 4 x^{4} + x \cdot 2 - x^{4} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.7

Move $2$ to the left of $x$ .

$f' (x) = \frac{x - 4 x^{4} + 2 \cdot x - x^{4} \cdot 2}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.8

Multiply $2$ by $- 1$ .

$f' (x) = \frac{x - 4 x^{4} + 2 \cdot x - 2 x^{4}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.9

Add $x$ and $2 x$ .

$f' (x) = \frac{- 4 x^{4} + 3 x - 2 x^{4}}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.10

Subtract $2 x^{4}$ from $- 4 x^{4}$ .

$f' (x) = \frac{- 6 x^{4} + 3 x}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.11

Factor $3 x$ out of $- 6 x^{4} + 3 x$ .

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Step 4.1.16.6.11.1

Factor $3 x$ out of $- 6 x^{4}$ .

$f' (x) = \frac{3 x (- 2 x^{3}) + 3 x}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.11.2

Factor $3 x$ out of $3 x$ .

$f' (x) = \frac{3 x (- 2 x^{3}) + 3 x (1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.6.11.3

Factor $3 x$ out of $3 x (- 2 x^{3}) + 3 x (1)$ .

$f' (x) = \frac{3 x (- 2 x^{3} + 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.7

Factor $- 1$ out of $- 2 x^{3}$ .

$f' (x) = \frac{3 x (- (2 x^{3}) + 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.8

Rewrite $1$ as $- 1 (- 1)$ .

$f' (x) = \frac{3 x (- (2 x^{3}) - 1 \cdot - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.9

Factor $- 1$ out of $- (2 x^{3}) - 1 (- 1)$ .

$f' (x) = \frac{3 x (- (2 x^{3} - 1))}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.10

Rewrite $- (2 x^{3} - 1)$ as $- 1 (2 x^{3} - 1)$ .

$f' (x) = \frac{3 x (- 1 (2 x^{3} - 1))}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.11

Move the negative in front of the fraction.

$f' (x) = - \frac{(3 x) (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.1.16.12

Reorder factors in $- \frac{(3 x) (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$f' (x) = - \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$ .

$- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$3 x (2 x^{3} - 1) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x^{3} - 1 = 0$

Step 5.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.3

Set $2 x^{3} - 1$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $2 x^{3} - 1$ equal to $0$ .

$2 x^{3} - 1 = 0$

Step 5.3.3.2

Solve $2 x^{3} - 1 = 0$ for $x$ .

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Step 5.3.3.2.1

Add $1$ to both sides of the equation.

$2 x^{3} = 1$

Step 5.3.3.2.2

Divide each term in $2 x^{3} = 1$ by $2$ and simplify.

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Step 5.3.3.2.2.1

Divide each term in $2 x^{3} = 1$ by $2$ .

$\frac{2 x^{3}}{2} = \frac{1}{2}$

Step 5.3.3.2.2.2

Simplify the left side.

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Step 5.3.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x^{3}}{2} = \frac{1}{2}$

Step 5.3.3.2.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{1}{2}$

Step 5.3.3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{1}{2}}$

Step 5.3.3.2.4

Simplify $\sqrt[3]{\frac{1}{2}}$ .

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Step 5.3.3.2.4.1

Rewrite $\sqrt[3]{\frac{1}{2}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$ .

$x = \frac{\sqrt[3]{1}}{\sqrt[3]{2}}$

Step 5.3.3.2.4.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[3]{2}}$

Step 5.3.3.2.4.3

Multiply $\frac{1}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{1}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 5.3.3.2.4.4

Combine and simplify the denominator.

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Step 5.3.3.2.4.4.1

Multiply $\frac{1}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 5.3.3.2.4.4.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 5.3.3.2.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 5.3.3.2.4.4.4

Add $1$ and $2$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 5.3.3.2.4.4.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 5.3.3.2.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 5.3.3.2.4.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 5.3.3.2.4.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 5.3.3.2.4.4.5.4

Cancel the common factor of $3$ .

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Step 5.3.3.2.4.4.5.4.1

Cancel the common factor.

$x = \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 5.3.3.2.4.4.5.4.2

Rewrite the expression.

$x = \frac{{\sqrt[3]{2}}^{2}}{2^{1}}$

Step 5.3.3.2.4.4.5.5

Evaluate the exponent.

$x = \frac{{\sqrt[3]{2}}^{2}}{2}$

Step 5.3.3.2.4.5

Simplify the numerator.

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Step 5.3.3.2.4.5.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$x = \frac{\sqrt[3]{2^{2}}}{2}$

Step 5.3.3.2.4.5.2

Raise $2$ to the power of $2$ .

$x = \frac{\sqrt[3]{4}}{2}$

Step 5.3.4

The final solution is all the values that make $3 x (2 x^{3} - 1) = 0$ true.

$x = 0, \frac{\sqrt[3]{4}}{2}$

Step 5.4

Exclude the solutions that do not make $- \frac{3 x (2 x^{3} - 1)}{2 {(x - x^{4})}^{\frac{1}{2}}} = 0$ true.

$x = \frac{\sqrt[3]{4}}{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{3 x (2 x^{3} - 1)}{2 \sqrt{{(x - x^{4})}^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$- \frac{3 x (2 x^{3} - 1)}{2 \sqrt{x - x^{4}}}$

Step 6.2

Set the denominator in $\frac{3 x (2 x^{3} - 1)}{2 \sqrt{x - x^{4}}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x - x^{4}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x - x^{4}})}^{2} = 0^{2}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - x^{4}}$ as ${(x - x^{4})}^{\frac{1}{2}}$ .

${(2 {(x - x^{4})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(2 {(x - x^{4})}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Apply the product rule to $2 {(x - x^{4})}^{\frac{1}{2}}$ .

$2^{2} {({(x - x^{4})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(x - x^{4})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2.1.3

Multiply the exponents in ${({(x - x^{4})}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(x - x^{4})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(x - x^{4})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(x - x^{4})}^{1} = 0^{2}$

Step 6.3.2.2.1.4

Simplify.

$4 (x - x^{4}) = 0^{2}$

Step 6.3.2.2.1.5

Apply the distributive property.

$4 x + 4 (- x^{4}) = 0^{2}$

Step 6.3.2.2.1.6

Multiply $- 1$ by $4$ .

$4 x - 4 x^{4} = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 x - 4 x^{4} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Factor the left side of the equation.

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Step 6.3.3.1.1

Factor $4 x$ out of $4 x - 4 x^{4}$ .

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Step 6.3.3.1.1.1

Factor $4 x$ out of $4 x$ .

$4 x (1) - 4 x^{4} = 0$

Step 6.3.3.1.1.2

Factor $4 x$ out of $- 4 x^{4}$ .

$4 x (1) + 4 x (- x^{3}) = 0$

Step 6.3.3.1.1.3

Factor $4 x$ out of $4 x (1) + 4 x (- x^{3})$ .

$4 x (1 - x^{3}) = 0$

Step 6.3.3.1.2

Rewrite $1$ as $1^{3}$ .

$4 x (1^{3} - x^{3}) = 0$

Step 6.3.3.1.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

$4 x ((1 - x) (1^{2} + 1 x + x^{2})) = 0$

Step 6.3.3.1.4

Factor.

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Step 6.3.3.1.4.1

Simplify.

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Step 6.3.3.1.4.1.1

One to any power is one.

$4 x ((1 - x) (1 + 1 x + x^{2})) = 0$

Step 6.3.3.1.4.1.2

Multiply $x$ by $1$ .

$4 x ((1 - x) (1 + x + x^{2})) = 0$

Step 6.3.3.1.4.2

Remove unnecessary parentheses.

$4 x (1 - x) (1 + x + x^{2}) = 0$

Step 6.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - x = 0$

$1 + x + x^{2} = 0$

Step 6.3.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.3.4

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 6.3.3.4.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 6.3.3.4.2

Solve $1 - x = 0$ for $x$ .

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Step 6.3.3.4.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 6.3.3.4.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 6.3.3.4.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 6.3.3.4.2.2.2

Simplify the left side.

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Step 6.3.3.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 6.3.3.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 6.3.3.4.2.2.3

Simplify the right side.

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Step 6.3.3.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 6.3.3.5

Set $1 + x + x^{2}$ equal to $0$ and solve for $x$ .

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Step 6.3.3.5.1

Set $1 + x + x^{2}$ equal to $0$ .

$1 + x + x^{2} = 0$

Step 6.3.3.5.2

Solve $1 + x + x^{2} = 0$ for $x$ .

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Step 6.3.3.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.3.3.5.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 6.3.3.5.2.3

Simplify.

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Step 6.3.3.5.2.3.1

Simplify the numerator.

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Step 6.3.3.5.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.3.3.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.3.3.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.3.3.5.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.3.3.5.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.3.3.5.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.3.3.5.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.3.3.5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.3.3.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.3.3.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.3.3.5.2.4.1

Simplify the numerator.

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Step 6.3.3.5.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.3.3.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.3.3.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.3.3.5.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.3.3.5.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.3.3.5.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.3.3.5.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.3.3.5.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.3.3.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.3.3.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{3}}{2}$

Step 6.3.3.5.2.4.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{3}}{2}$

Step 6.3.3.5.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{2}$

Step 6.3.3.5.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$x = \frac{- 1 (1 - i \sqrt{3})}{2}$

Step 6.3.3.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{3}}{2}$

Step 6.3.3.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.3.3.5.2.5.1

Simplify the numerator.

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Step 6.3.3.5.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.3.3.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.3.3.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.3.3.5.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.3.3.5.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.3.3.5.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.3.3.5.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.3.3.5.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.3.3.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.3.3.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{3}}{2}$

Step 6.3.3.5.2.5.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{3}}{2}$

Step 6.3.3.5.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{3})}{2}$

Step 6.3.3.5.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$x = \frac{- 1 (1 + i \sqrt{3})}{2}$

Step 6.3.3.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{3}}{2}$

Step 6.3.3.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 6.3.3.6

The final solution is all the values that make $4 x (1 - x) (1 + x + x^{2}) = 0$ true.

$x = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 6.4

Set the radicand in $\sqrt{x - x^{4}}$ less than $0$ to find where the expression is undefined.

$x - x^{4} < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Convert the inequality to an equation.

$x - x^{4} = 0$

Step 6.5.2

Factor the left side of the equation.

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Step 6.5.2.1

Factor $x$ out of $x - x^{4}$ .

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Step 6.5.2.1.1

Raise $x$ to the power of $1$ .

$x - x^{4} = 0$

Step 6.5.2.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{4} = 0$

Step 6.5.2.1.3

Factor $x$ out of $- x^{4}$ .

$x \cdot 1 + x (- x^{3}) = 0$

Step 6.5.2.1.4

Factor $x$ out of $x \cdot 1 + x (- x^{3})$ .

$x (1 - x^{3}) = 0$

Step 6.5.2.2

Rewrite $1$ as $1^{3}$ .

$x (1^{3} - x^{3}) = 0$

Step 6.5.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

$x ((1 - x) (1^{2} + 1 x + x^{2})) = 0$

Step 6.5.2.4

Factor.

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Step 6.5.2.4.1

Simplify.

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Step 6.5.2.4.1.1

One to any power is one.

$x ((1 - x) (1 + 1 x + x^{2})) = 0$

Step 6.5.2.4.1.2

Multiply $x$ by $1$ .

$x ((1 - x) (1 + x + x^{2})) = 0$

Step 6.5.2.4.2

Remove unnecessary parentheses.

$x (1 - x) (1 + x + x^{2}) = 0$

Step 6.5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - x = 0$

$1 + x + x^{2} = 0$

Step 6.5.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5.5

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 6.5.5.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 6.5.5.2

Solve $1 - x = 0$ for $x$ .

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Step 6.5.5.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 6.5.5.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 6.5.5.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 6.5.5.2.2.2

Simplify the left side.

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Step 6.5.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 6.5.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 6.5.5.2.2.3

Simplify the right side.

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Step 6.5.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 6.5.6

Set $1 + x + x^{2}$ equal to $0$ and solve for $x$ .

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Step 6.5.6.1

Set $1 + x + x^{2}$ equal to $0$ .

$1 + x + x^{2} = 0$

Step 6.5.6.2

Solve $1 + x + x^{2} = 0$ for $x$ .

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Step 6.5.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.5.6.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 6.5.6.2.3

Simplify.

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Step 6.5.6.2.3.1

Simplify the numerator.

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Step 6.5.6.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.5.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.5.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.5.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.5.6.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.5.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.5.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.5.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.5.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.5.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.5.6.2.4.1

Simplify the numerator.

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Step 6.5.6.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.5.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.5.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.5.6.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.5.6.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.5.6.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.5.6.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.5.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.5.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.5.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{3}}{2}$

Step 6.5.6.2.4.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{3}}{2}$

Step 6.5.6.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{2}$

Step 6.5.6.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$x = \frac{- 1 (1 - i \sqrt{3})}{2}$

Step 6.5.6.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{3}}{2}$

Step 6.5.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.5.6.2.5.1

Simplify the numerator.

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Step 6.5.6.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.5.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.5.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.5.6.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.5.6.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.5.6.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.5.6.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.5.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.5.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.5.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{3}}{2}$

Step 6.5.6.2.5.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{3}}{2}$

Step 6.5.6.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{3})}{2}$

Step 6.5.6.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$x = \frac{- 1 (1 + i \sqrt{3})}{2}$

Step 6.5.6.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{3}}{2}$

Step 6.5.6.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 6.5.7

The final solution is all the values that make $x (1 - x) (1 + x + x^{2}) = 0$ true.

$x = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 6.5.8

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 6.5.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.5.9.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 6.5.9.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 6.5.9.1.2

Replace $x$ with $- 2$ in the original inequality.

$(- 2) - {(- 2)}^{4} < 0$

Step 6.5.9.1.3

The left side $- 18$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.5.9.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 6.5.9.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 6.5.9.2.2

Replace $x$ with $0.5$ in the original inequality.

$(0.5) - {(0.5)}^{4} < 0$

Step 6.5.9.2.3

The left side $0.4375$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.5.9.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 6.5.9.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 6.5.9.3.2

Replace $x$ with $4$ in the original inequality.

$(4) - {(4)}^{4} < 0$

Step 6.5.9.3.3

The left side $- 252$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.5.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ True

$0 < x < 1$ False

$x > 1$ True

$x < 0$ True

$0 < x < 1$ False

$x > 1$ True

Step 6.5.10

The solution consists of all of the true intervals.

$x < 0$ or $x > 1$

Step 6.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0, x \geq 1$

$(- \infty, 0] \cup [1, \infty)$

$x \leq 0, x \geq 1$

$(- \infty, 0] \cup [1, \infty)$

Step 7

Critical points to evaluate.

$x = \frac{\sqrt[3]{4}}{2}, 0, 1$

Step 8

Evaluate the second derivative at $x = \frac{\sqrt[3]{4}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 (8 {(\frac{\sqrt[3]{4}}{2})}^{6} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - (\frac{\sqrt[3]{4}}{2})) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9

Evaluate the second derivative.

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Step 9.1

Remove parentheses.

Step 9.2

Simplify the numerator.

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Step 9.2.1

Apply the product rule to $\frac{\sqrt[3]{4}}{2}$ .

$\frac{3 (8 \frac{{\sqrt[3]{4}}^{6}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

Rewrite ${\sqrt[3]{4}}^{6}$ as $4^{2}$ .

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Step 9.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{3 (8 \frac{{(4^{\frac{1}{3}})}^{6}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (8 \frac{4^{\frac{1}{3} \cdot 6}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.3

Combine $\frac{1}{3}$ and $6$ .

$\frac{3 (8 \frac{4^{\frac{6}{3}}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.4

Cancel the common factor of $6$ and $3$ .

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Step 9.2.2.1.4.1

Factor $3$ out of $6$ .

$\frac{3 (8 \frac{4^{\frac{3 \cdot 2}{3}}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.4.2

Cancel the common factors.

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Step 9.2.2.1.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (8 \frac{4^{\frac{3 \cdot 2}{3 (1)}}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.4.2.2

Cancel the common factor.

$\frac{3 (8 \frac{4^{\frac{3 \cdot 2}{3 \cdot 1}}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.4.2.3

Rewrite the expression.

$\frac{3 (8 \frac{4^{\frac{2}{1}}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.1.4.2.4

Divide $2$ by $1$ .

$\frac{3 (8 \frac{4^{2}}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.2.2

Raise $4$ to the power of $2$ .

$\frac{3 (8 \frac{16}{2^{6}} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.3

Raise $2$ to the power of $6$ .

$\frac{3 (8 (\frac{16}{64}) - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.4

Cancel the common factor of $8$ .

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Step 9.2.4.1

Factor $8$ out of $64$ .

$\frac{3 (8 \frac{16}{8 (8)} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.4.2

Cancel the common factor.

$\frac{3 (8 \frac{16}{8 \cdot 8} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.4.3

Rewrite the expression.

$\frac{3 (\frac{16}{8} - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.5

Divide $16$ by $8$ .

$\frac{3 (2 - 12 {(\frac{\sqrt[3]{4}}{2})}^{3} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.6

Apply the product rule to $\frac{\sqrt[3]{4}}{2}$ .

$\frac{3 (2 - 12 \frac{{\sqrt[3]{4}}^{3}}{2^{3}} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.7

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 9.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{3 (2 - 12 \frac{{(4^{\frac{1}{3}})}^{3}}{2^{3}} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (2 - 12 \frac{4^{\frac{1}{3} \cdot 3}}{2^{3}} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.7.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{3 (2 - 12 \frac{4^{\frac{3}{3}}}{2^{3}} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.7.4

Cancel the common factor of $3$ .

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Step 9.2.7.4.1

Cancel the common factor.

Step 9.2.7.4.2

Rewrite the expression.

$\frac{3 (2 - 12 \frac{4^{1}}{2^{3}} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.7.5

Evaluate the exponent.

$\frac{3 (2 - 12 \frac{4}{2^{3}} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.8

Raise $2$ to the power of $3$ .

$\frac{3 (2 - 12 (\frac{4}{8}) + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.9

Cancel the common factor of $4$ .

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Step 9.2.9.1

Factor $4$ out of $- 12$ .

$\frac{3 (2 + 4 (- 3) \frac{4}{8} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.9.2

Factor $4$ out of $8$ .

$\frac{3 (2 + 4 \cdot - 3 \frac{4}{4 \cdot 2} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.9.3

Cancel the common factor.

Step 9.2.9.4

Rewrite the expression.

$\frac{3 (2 - 3 (\frac{4}{2}) + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.10

Combine $- 3$ and $\frac{4}{2}$ .

$\frac{3 (2 + \frac{- 3 \cdot 4}{2} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.11

Multiply $- 3$ by $4$ .

$\frac{3 (2 + \frac{- 12}{2} + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.12

Divide $- 12$ by $2$ .

$\frac{3 (2 - 6 + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.13

Subtract $6$ from $2$ .

$\frac{3 (- 4 + 1)}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.2.14

Add $- 4$ and $1$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + {(\frac{\sqrt[3]{4}}{2})}^{2})}$

Step 9.3

Simplify the denominator.

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Step 9.3.1

Apply the product rule to $\frac{\sqrt[3]{4}}{2}$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{{\sqrt[3]{4}}^{2}}{2^{2}})}$

Step 9.3.2

Simplify the numerator.

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Step 9.3.2.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{4^{2}}}{2^{2}})}$

Step 9.3.2.2

Raise $4$ to the power of $2$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{16}}{2^{2}})}$

Step 9.3.2.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 9.3.2.3.1

Factor $8$ out of $16$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{8 (2)}}{2^{2}})}$

Step 9.3.2.3.2

Rewrite $8$ as $2^{3}$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2^{3} \cdot 2}}{2^{2}})}$

Step 9.3.2.4

Pull terms out from under the radical.

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{2 \sqrt[3]{2}}{2^{2}})}$

Step 9.3.3

Raise $2$ to the power of $2$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{2 \sqrt[3]{2}}{4})}$

Step 9.3.4

Cancel the common factor of $2$ and $4$ .

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Step 9.3.4.1

Factor $2$ out of $2 \sqrt[3]{2}$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{2 (\sqrt[3]{2})}{4})}$

Step 9.3.4.2

Cancel the common factors.

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Step 9.3.4.2.1

Factor $2$ out of $4$ .

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{2 \sqrt[3]{2}}{2 \cdot 2})}$

Step 9.3.4.2.2

Cancel the common factor.

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{2 \sqrt[3]{2}}{2 \cdot 2})}$

Step 9.3.4.2.3

Rewrite the expression.

$\frac{3 \cdot - 3}{4 {(- {(\frac{\sqrt[3]{4}}{2})}^{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5

Simplify each term.

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Step 9.3.5.1

Apply the product rule to $\frac{\sqrt[3]{4}}{2}$ .

$\frac{3 \cdot - 3}{4 {(- \frac{{\sqrt[3]{4}}^{4}}{2^{4}} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.2

Simplify the numerator.

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Step 9.3.5.2.1

Rewrite ${\sqrt[3]{4}}^{4}$ as $\sqrt[3]{4^{4}}$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{4^{4}}}{2^{4}} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.2.2

Raise $4$ to the power of $4$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{256}}{2^{4}} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.2.3

Rewrite $256$ as $4^{3} \cdot 4$ .

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Step 9.3.5.2.3.1

Factor $64$ out of $256$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{64 (4)}}{2^{4}} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.2.3.2

Rewrite $64$ as $4^{3}$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{4^{3} \cdot 4}}{2^{4}} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.2.4

Pull terms out from under the radical.

$\frac{3 \cdot - 3}{4 {(- \frac{4 \sqrt[3]{4}}{2^{4}} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.3

Raise $2$ to the power of $4$ .

$\frac{3 \cdot - 3}{4 {(- \frac{4 \sqrt[3]{4}}{16} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.4

Cancel the common factor of $4$ and $16$ .

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Step 9.3.5.4.1

Factor $4$ out of $4 \sqrt[3]{4}$ .

$\frac{3 \cdot - 3}{4 {(- \frac{4 (\sqrt[3]{4})}{16} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.4.2

Cancel the common factors.

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Step 9.3.5.4.2.1

Factor $4$ out of $16$ .

$\frac{3 \cdot - 3}{4 {(- \frac{4 \sqrt[3]{4}}{4 \cdot 4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.4.2.2

Cancel the common factor.

$\frac{3 \cdot - 3}{4 {(- \frac{4 \sqrt[3]{4}}{4 \cdot 4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.5.4.2.3

Rewrite the expression.

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{4}}{4} + \frac{\sqrt[3]{4}}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.6

To write $\frac{\sqrt[3]{4}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{4}}{4} + \frac{\sqrt[3]{4}}{2} \cdot \frac{2}{2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.7.1

Multiply $\frac{\sqrt[3]{4}}{2}$ by $\frac{2}{2}$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{4}}{4} + \frac{\sqrt[3]{4} \cdot 2}{2 \cdot 2})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.7.2

Multiply $2$ by $2$ .

$\frac{3 \cdot - 3}{4 {(- \frac{\sqrt[3]{4}}{4} + \frac{\sqrt[3]{4} \cdot 2}{4})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.8

Combine the numerators over the common denominator.

$\frac{3 \cdot - 3}{4 {(\frac{- \sqrt[3]{4} + \sqrt[3]{4} \cdot 2}{4})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.9

Simplify the numerator.

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Step 9.3.9.1

Move $2$ to the left of $\sqrt[3]{4}$ .

$\frac{3 \cdot - 3}{4 {(\frac{- \sqrt[3]{4} + 2 \cdot \sqrt[3]{4}}{4})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.9.2

Add $- \sqrt[3]{4}$ and $2 \sqrt[3]{4}$ .

$\frac{3 \cdot - 3}{4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (1 - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.10

Write $1$ as a fraction with a common denominator.

$\frac{3 \cdot - 3}{4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (\frac{2}{2} - \frac{\sqrt[3]{4}}{2}) (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.11

Combine the numerators over the common denominator.

$\frac{3 \cdot - 3}{4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} \frac{2 - \sqrt[3]{4}}{2} (1 + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.12

Write $1$ as a fraction with a common denominator.

$\frac{3 \cdot - 3}{4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} \frac{2 - \sqrt[3]{4}}{2} (\frac{2}{2} + \frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.13

Combine the numerators over the common denominator.

$\frac{3 \cdot - 3}{4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} \frac{2 - \sqrt[3]{4}}{2} (\frac{2 + \sqrt[3]{4}}{2} + \frac{\sqrt[3]{2}}{2})}$

Step 9.3.14

Combine the numerators over the common denominator.

$\frac{3 \cdot - 3}{4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} \frac{2 - \sqrt[3]{4}}{2} \frac{2 + \sqrt[3]{4} + \sqrt[3]{2}}{2}}$

Step 9.3.15

Combine exponents.

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Step 9.3.15.1

Combine $\frac{2 - \sqrt[3]{4}}{2}$ and $4$ .

$\frac{3 \cdot - 3}{\frac{(2 - \sqrt[3]{4}) \cdot 4}{2} {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} \frac{2 + \sqrt[3]{4} + \sqrt[3]{2}}{2}}$

Step 9.3.15.2

Combine $\frac{(2 - \sqrt[3]{4}) \cdot 4}{2}$ and ${(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}}$ .

$\frac{3 \cdot - 3}{\frac{(2 - \sqrt[3]{4}) \cdot 4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}}}{2} \cdot \frac{2 + \sqrt[3]{4} + \sqrt[3]{2}}{2}}$

Step 9.3.15.3

Multiply $\frac{(2 - \sqrt[3]{4}) \cdot 4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}}}{2}$ by $\frac{2 + \sqrt[3]{4} + \sqrt[3]{2}}{2}$ .

$\frac{3 \cdot - 3}{\frac{(2 - \sqrt[3]{4}) \cdot 4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}{2 \cdot 2}}$

Step 9.3.15.4

Multiply $2$ by $2$ .

$\frac{3 \cdot - 3}{\frac{(2 - \sqrt[3]{4}) \cdot 4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}{4}}$

Step 9.3.16

Reduce the expression by cancelling the common factors.

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Step 9.3.16.1

Reduce the expression $\frac{(2 - \sqrt[3]{4}) \cdot 4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}{4}$ by cancelling the common factors.

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Step 9.3.16.1.1

Cancel the common factor.

$\frac{3 \cdot - 3}{\frac{(2 - \sqrt[3]{4}) \cdot 4 {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}{4}}$

Step 9.3.16.1.2

Rewrite the expression.

$\frac{3 \cdot - 3}{\frac{(2 - \sqrt[3]{4}) {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}{1}}$

Step 9.3.16.2

Divide $(2 - \sqrt[3]{4}) {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})$ by $1$ .

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) {(\frac{\sqrt[3]{4}}{4})}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.3.17

Apply the product rule to $\frac{\sqrt[3]{4}}{4}$ .

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{4^{\frac{1}{2}}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.3.18

Simplify the denominator.

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Step 9.3.18.1

Rewrite $4$ as $2^{2}$ .

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{{(2^{2})}^{\frac{1}{2}}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.3.18.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2^{2 (\frac{1}{2})}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.3.18.3

Cancel the common factor of $2$ .

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Step 9.3.18.3.1

Cancel the common factor.

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2^{2 (\frac{1}{2})}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.3.18.3.2

Rewrite the expression.

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2^{1}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.3.18.4

Evaluate the exponent.

$\frac{3 \cdot - 3}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.4

Multiply $3$ by $- 3$ .

$\frac{- 9}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.5

Factor $\frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2}$ out of $\frac{- 9}{(2 - \sqrt[3]{4}) \frac{{\sqrt[3]{4}}^{\frac{1}{2}}}{2} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$ .

$\frac{2}{{\sqrt[3]{4}}^{\frac{1}{2}}} \cdot \frac{- 9}{(2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.6

Combine.

$\frac{2 \cdot - 9}{{\sqrt[3]{4}}^{\frac{1}{2}} ((2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2}))}$

Step 9.7

Simplify the expression.

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Step 9.7.1

Multiply $2$ by $- 9$ .

$\frac{- 18}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.7.2

Move the negative in front of the fraction.

$- \frac{18}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.8

Multiply $\frac{18}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2})}$ by $\frac{2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}}{2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}}$ .

$- (\frac{18}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2})} \cdot \frac{2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}}{2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}})$

Step 9.9

$- \frac{18 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 - \sqrt[3]{4}) (2 + \sqrt[3]{4} + \sqrt[3]{2}) (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}$

Step 9.10

Reorder.

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Step 9.10.1

Move $2 - \sqrt[3]{4}$ .

$- \frac{18 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) ((2 - \sqrt[3]{4}) (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}))}$

Step 9.10.2

Expand the denominator using the FOIL method.

$- \frac{18 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) (2^{3} + 4 \sqrt[3]{4} + 2 {(- \sqrt[3]{4})}^{2} - \sqrt[3]{4} \cdot 2^{2} - 2 {\sqrt[3]{4}}^{2} - \sqrt[3]{4} {(- \sqrt[3]{4})}^{2})}$

Step 9.10.3

Simplify.

$- \frac{18 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 4}$

Step 9.11

Factor $2$ out of $18 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})$ .

$- \frac{2 (9 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}))}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 4}$

Step 9.12

Cancel the common factors.

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Step 9.12.1

Factor $2$ out of ${\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 4$ .

$- \frac{2 (9 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}))}{2 ({\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2)}$

Step 9.12.2

Cancel the common factor.

$- \frac{2 (9 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2}))}{2 ({\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2)}$

Step 9.12.3

Rewrite the expression.

$- \frac{9 (2^{2} - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13

Simplify the numerator.

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Step 9.13.1

Raise $2$ to the power of $2$ .

$- \frac{9 (4 - 2 (- \sqrt[3]{4}) + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.2

Multiply $- 1$ by $- 2$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + {(- \sqrt[3]{4})}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.3

Apply the product rule to $- \sqrt[3]{4}$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + {(- 1)}^{2} {\sqrt[3]{4}}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.4

Raise $- 1$ to the power of $2$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + 1 {\sqrt[3]{4}}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.5

Multiply ${\sqrt[3]{4}}^{2}$ by $1$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + {\sqrt[3]{4}}^{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.6

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + \sqrt[3]{4^{2}})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.7

Raise $4$ to the power of $2$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + \sqrt[3]{16})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.8

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 9.13.8.1

Factor $8$ out of $16$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + \sqrt[3]{8 (2)})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.8.2

Rewrite $8$ as $2^{3}$ .

$- \frac{9 (4 + 2 \sqrt[3]{4} + \sqrt[3]{2^{3} \cdot 2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.13.9

Pull terms out from under the radical.

$- \frac{9 (4 + 2 \sqrt[3]{4} + 2 \sqrt[3]{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.14

Factor $2$ out of $9 (4 + 2 \sqrt[3]{4} + 2 \sqrt[3]{2})$ .

$- \frac{2 (9 (2 + \sqrt[3]{4} + \sqrt[3]{2}))}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2}$

Step 9.15

Cancel the common factors.

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Step 9.15.1

Factor $2$ out of ${\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}) \cdot 2$ .

$- \frac{2 (9 (2 + \sqrt[3]{4} + \sqrt[3]{2}))}{2 \cdot ({\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}))}$

Step 9.15.2

Cancel the common factor.

$- \frac{2 (9 (2 + \sqrt[3]{4} + \sqrt[3]{2}))}{2 \cdot ({\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2}))}$

Step 9.15.3

Rewrite the expression.

$- \frac{9 (2 + \sqrt[3]{4} + \sqrt[3]{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.16

Cancel the common factor.

$- \frac{9 (2 + \sqrt[3]{4} + \sqrt[3]{2})}{{\sqrt[3]{4}}^{\frac{1}{2}} (2 + \sqrt[3]{4} + \sqrt[3]{2})}$

Step 9.17

Rewrite the expression.

$- \frac{9}{{\sqrt[3]{4}}^{\frac{1}{2}}}$

Step 10

$x = \frac{\sqrt[3]{4}}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{\sqrt[3]{4}}{2}$ is a local maximum

Step 11

Find the y-value when $x = \frac{\sqrt[3]{4}}{2}$ .

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Step 11.1

Replace the variable $x$ with $\frac{\sqrt[3]{4}}{2}$ in the expression.

$f (\frac{\sqrt[3]{4}}{2}) = (\frac{\sqrt[3]{4}}{2}) \sqrt{(\frac{\sqrt[3]{4}}{2}) - {(\frac{\sqrt[3]{4}}{2})}^{4}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Apply the product rule to $\frac{\sqrt[3]{4}}{2}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{{\sqrt[3]{4}}^{4}}{2^{4}}}$

Step 11.2.2

Simplify the numerator.

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Step 11.2.2.1

Rewrite ${\sqrt[3]{4}}^{4}$ as $\sqrt[3]{4^{4}}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{\sqrt[3]{4^{4}}}{2^{4}}}$

Step 11.2.2.2

Raise $4$ to the power of $4$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{\sqrt[3]{256}}{2^{4}}}$

Step 11.2.2.3

Rewrite $256$ as $4^{3} \cdot 4$ .

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Step 11.2.2.3.1

Factor $64$ out of $256$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{\sqrt[3]{64 (4)}}{2^{4}}}$

Step 11.2.2.3.2

Rewrite $64$ as $4^{3}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{\sqrt[3]{4^{3} \cdot 4}}{2^{4}}}$

Step 11.2.2.4

Pull terms out from under the radical.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{4 \sqrt[3]{4}}{2^{4}}}$

Step 11.2.3

Reduce the expression by cancelling the common factors.

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Step 11.2.3.1

Raise $2$ to the power of $4$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{4 \sqrt[3]{4}}{16}}$

Step 11.2.3.2

Cancel the common factor of $4$ and $16$ .

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Step 11.2.3.2.1

Factor $4$ out of $4 \sqrt[3]{4}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{4 (\sqrt[3]{4})}{16}}$

Step 11.2.3.2.2

Cancel the common factors.

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Step 11.2.3.2.2.1

Factor $4$ out of $16$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{4 \sqrt[3]{4}}{4 \cdot 4}}$

Step 11.2.3.2.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{4 \sqrt[3]{4}}{4 \cdot 4}}$

Step 11.2.3.2.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} - \frac{\sqrt[3]{4}}{4}}$

Step 11.2.4

To write $\frac{\sqrt[3]{4}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{2} \cdot \frac{2}{2} - \frac{\sqrt[3]{4}}{4}}$

Step 11.2.5

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.5.1

Multiply $\frac{\sqrt[3]{4}}{2}$ by $\frac{2}{2}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4} \cdot 2}{2 \cdot 2} - \frac{\sqrt[3]{4}}{4}}$

Step 11.2.5.2

Multiply $2$ by $2$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4} \cdot 2}{4} - \frac{\sqrt[3]{4}}{4}}$

Step 11.2.6

Combine the numerators over the common denominator.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4} \cdot 2 - \sqrt[3]{4}}{4}}$

Step 11.2.7

Rewrite $\frac{\sqrt[3]{4} \cdot 2 - \sqrt[3]{4}}{4}$ in a factored form.

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Step 11.2.7.1

Move $2$ to the left of $\sqrt[3]{4}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{2 \cdot \sqrt[3]{4} - \sqrt[3]{4}}{4}}$

Step 11.2.7.2

Subtract $\sqrt[3]{4}$ from $2 \sqrt[3]{4}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \sqrt{\frac{\sqrt[3]{4}}{4}}$

Step 11.2.8

Rewrite $\sqrt{\frac{\sqrt[3]{4}}{4}}$ as $\frac{\sqrt{\sqrt[3]{4}}}{\sqrt{4}}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt{\sqrt[3]{4}}}{\sqrt{4}}$

Step 11.2.9

Simplify the numerator.

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Step 11.2.9.1

Rewrite $\sqrt{\sqrt[3]{4}}$ as $\sqrt[6]{4}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[6]{4}}{\sqrt{4}}$

Step 11.2.9.2

Rewrite $4$ as $2^{2}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[6]{2^{2}}}{\sqrt{4}}$

Step 11.2.9.3

Rewrite $\sqrt[6]{2^{2}}$ as $\sqrt[3]{\sqrt{2^{2}}}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[3]{\sqrt{2^{2}}}}{\sqrt{4}}$

Step 11.2.9.4

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[3]{2}}{\sqrt{4}}$

Step 11.2.10

Simplify the denominator.

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Step 11.2.10.1

Rewrite $4$ as $2^{2}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[3]{2}}{\sqrt{2^{2}}}$

Step 11.2.10.2

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[3]{2}}{2}$

Step 11.2.11

Multiply $\frac{\sqrt[3]{4}}{2} \cdot \frac{\sqrt[3]{2}}{2}$ .

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Step 11.2.11.1

Multiply $\frac{\sqrt[3]{4}}{2}$ by $\frac{\sqrt[3]{2}}{2}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4} \sqrt[3]{2}}{2 \cdot 2}$

Step 11.2.11.2

Combine using the product rule for radicals.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{4 \cdot 2}}{2 \cdot 2}$

Step 11.2.11.3

Multiply $4$ by $2$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{8}}{2 \cdot 2}$

Step 11.2.11.4

Multiply $2$ by $2$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{8}}{4}$

Step 11.2.12

Simplify the numerator.

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Step 11.2.12.1

Rewrite $8$ as $2^{3}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{2^{3}}}{4}$

Step 11.2.12.2

Pull terms out from under the radical, assuming real numbers.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{2}{4}$

Step 11.2.13

Cancel the common factor of $2$ and $4$ .

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Step 11.2.13.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{2 (1)}{4}$

Step 11.2.13.2

Cancel the common factors.

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Step 11.2.13.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{2 \cdot 1}{2 \cdot 2}$

Step 11.2.13.2.2

Cancel the common factor.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{2 \cdot 1}{2 \cdot 2}$

Step 11.2.13.2.3

Rewrite the expression.

$f (\frac{\sqrt[3]{4}}{2}) = \frac{1}{2}$

Step 11.2.14

The final answer is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 {(- {(0)}^{4} + 0)}^{\frac{1}{2}} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13

Evaluate the second derivative.

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Step 13.1

Remove parentheses.

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 {(- {(0)}^{4} + 0)}^{\frac{1}{2}} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.2

Simplify each term.

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Step 13.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 {(- 0 + 0)}^{\frac{1}{2}} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.2.2

Multiply $- 1$ by $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 {(0 + 0)}^{\frac{1}{2}} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3

Reduce the expression by cancelling the common factors.

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Step 13.3.1

Add $0$ and $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 \cdot 0^{\frac{1}{2}} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.2

Simplify the expression.

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Step 13.3.2.1

Rewrite $0$ as $0^{2}$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 \cdot {(0^{2})}^{\frac{1}{2}} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 \cdot 0^{2 (\frac{1}{2})} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.3

Cancel the common factor of $2$ .

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Step 13.3.3.1

Cancel the common factor.

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 \cdot 0^{2 (\frac{1}{2})} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.3.2

Rewrite the expression.

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 \cdot 0^{1} (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.4

Simplify the expression.

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Step 13.3.4.1

Evaluate the exponent.

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{4 \cdot 0 (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.4.2

Multiply $4$ by $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0 (1 - (0)) (1 + 0 + {(0)}^{2})}$

Step 13.3.4.3

Subtract $0$ from $1$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0 \cdot 1 (1 + 0 + {(0)}^{2})}$

Step 13.3.4.4

Multiply $0$ by $1$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0 (1 + 0 + {(0)}^{2})}$

Step 13.3.4.5

Raising $0$ to any positive power yields $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0 (1 + 0 + 0)}$

Step 13.3.4.6

Add $1$ and $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0 (1 + 0)}$

Step 13.3.4.7

Add $1$ and $0$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0 \cdot 1}$

Step 13.3.4.8

Multiply $0$ by $1$ .

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0}$

Step 13.3.4.9

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0}$

Step 13.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{3 (8 {(0)}^{6} - 12 {(0)}^{3} + 1)}{0}$

Step 13.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15