Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.2.2

Apply the product rule to $2 (x)$ .

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

Step 1.3

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

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

Step 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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = {(1 - 4 x)}^{3}$ .

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u$ as $1 - 4 x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $1 - 4 x$ .

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

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $1 - 4 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x]$ .

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

Step 4.2

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (3 {(1 - 4 x)}^{2} (0 + \frac{d}{d x} [- 4 x])) + {(1 - 4 x)}^{3} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.3

Add $0$ and $\frac{d}{d x} [- 4 x]$ .

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

Step 4.4

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

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

Step 4.5

Multiply $- 4$ by $3$ .

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2} \frac{d}{d x} [x]) + {(1 - 4 x)}^{3} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.6

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2} \cdot 1) + {(1 - 4 x)}^{3} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.7

Multiply $- 12$ by $1$ .

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.8

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 5

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 6

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 7

Combine the numerators over the common denominator.

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 8.2

Subtract $2$ from $1$ .

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 9

Move the negative in front of the fraction.

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 10

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + {(1 - 4 x)}^{3} \frac{x^{- \frac{1}{2}}}{2})$

Step 11

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + \frac{{(1 - 4 x)}^{3} x^{- \frac{1}{2}}}{2})$

Step 12

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

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) + \frac{{(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}})$

Step 13

To write $x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2})$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$2^{\frac{1}{2}} (x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{{(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}})$

Step 14

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

$2^{\frac{1}{2}} (\frac{x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} + \frac{{(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}})$

Step 15

Combine the numerators over the common denominator.

$2^{\frac{1}{2}} \frac{x^{\frac{1}{2}} (- 12 {(1 - 4 x)}^{2}) (2 x^{\frac{1}{2}}) + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 16

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

$2^{\frac{1}{2}} \frac{- 12 {(1 - 4 x)}^{2} (2 x^{\frac{1}{2} + \frac{1}{2}}) + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 17

Simplify the expression.

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Step 17.1

Combine the numerators over the common denominator.

$2^{\frac{1}{2}} \frac{- 12 {(1 - 4 x)}^{2} (2 x^{\frac{1 + 1}{2}}) + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 17.2

Add $1$ and $1$ .

$2^{\frac{1}{2}} \frac{- 12 {(1 - 4 x)}^{2} (2 x^{\frac{2}{2}}) + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 18

Cancel the common factor of $2$ .

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Step 18.1

Cancel the common factor.

$2^{\frac{1}{2}} \frac{- 12 {(1 - 4 x)}^{2} (2 x^{\frac{2}{2}}) + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 18.2

Rewrite the expression.

$2^{\frac{1}{2}} \frac{- 12 {(1 - 4 x)}^{2} (2 x^{1}) + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 19

Multiply $2$ by $- 12$ .

$2^{\frac{1}{2}} \frac{- 24 {(1 - 4 x)}^{2} x^{1} + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 20

Simplify.

$2^{\frac{1}{2}} \frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$

Step 21

Combine $2^{\frac{1}{2}}$ and $\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}}}$ .

$\frac{2^{\frac{1}{2}} (- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3})}{2 x^{\frac{1}{2}}}$

Step 22

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

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2 x^{\frac{1}{2}} \cdot 2^{- \frac{1}{2}}}$

Step 23

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

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Step 23.1

Move $2^{- \frac{1}{2}}$ .

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2^{- \frac{1}{2}} \cdot 2 x^{\frac{1}{2}}}$

Step 23.2

Multiply $2^{- \frac{1}{2}}$ by $2$ .

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Step 23.2.1

Raise $2$ to the power of $1$ .

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2^{- \frac{1}{2}} \cdot 2^{1} x^{\frac{1}{2}}}$

Step 23.2.2

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

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2^{- \frac{1}{2} + 1} x^{\frac{1}{2}}}$

Step 23.3

Write $1$ as a fraction with a common denominator.

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2^{- \frac{1}{2} + \frac{2}{2}} x^{\frac{1}{2}}}$

Step 23.4

Combine the numerators over the common denominator.

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2^{\frac{- 1 + 2}{2}} x^{\frac{1}{2}}}$

Step 23.5

Add $- 1$ and $2$ .

$\frac{- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24

Simplify.

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Step 24.1

Simplify the numerator.

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Step 24.1.1

Factor ${(1 - 4 x)}^{2}$ out of $- 24 {(1 - 4 x)}^{2} x + {(1 - 4 x)}^{3}$ .

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Step 24.1.1.1

Factor ${(1 - 4 x)}^{2}$ out of $- 24 {(1 - 4 x)}^{2} x$ .

$\frac{{(1 - 4 x)}^{2} (- 24 x) + {(1 - 4 x)}^{3}}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.1.1.2

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

$\frac{{(1 - 4 x)}^{2} (- 24 x) + {(1 - 4 x)}^{2} (1 - 4 x)}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.1.1.3

Factor ${(1 - 4 x)}^{2}$ out of ${(1 - 4 x)}^{2} (- 24 x) + {(1 - 4 x)}^{2} (1 - 4 x)$ .

$\frac{{(1 - 4 x)}^{2} (- 24 x + 1 - 4 x)}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.1.2

Subtract $4 x$ from $- 24 x$ .

$\frac{{(1 - 4 x)}^{2} (- 28 x + 1)}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.2

Factor $- 1$ out of $- 28 x$ .

$\frac{{(1 - 4 x)}^{2} (- (28 x) + 1)}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.3

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

$\frac{{(1 - 4 x)}^{2} (- (28 x) - 1 (- 1))}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.4

Factor $- 1$ out of $- (28 x) - 1 (- 1)$ .

$\frac{{(1 - 4 x)}^{2} (- (28 x - 1))}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.5

Rewrite $- (28 x - 1)$ as $- 1 (28 x - 1)$ .

$\frac{{(1 - 4 x)}^{2} (- 1 (28 x - 1))}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 24.6

Move the negative in front of the fraction.

$- \frac{{(1 - 4 x)}^{2} (28 x - 1)}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$