Calculus Examples

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

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

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

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

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

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 $g (x) = 4 - x^{2}$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.9

By the Sum Rule, the derivative of $4 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

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

Step 1.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 1.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

Combine fractions.

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Step 1.14.1

Multiply $2$ by $- 1$ .

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

Step 1.14.2

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

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

Step 1.14.3

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

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

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

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

Step 1.18

Add $1$ and $1$ .

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

Step 1.19

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

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

Step 1.20

Cancel the common factors.

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Step 1.20.1

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

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

Step 1.20.2

Cancel the common factor.

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

Step 1.20.3

Rewrite the expression.

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

Step 1.21

Move the negative in front of the fraction.

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

Step 1.22

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

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

Step 1.23

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

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

Step 1.24

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

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

Step 1.25

Combine the numerators over the common denominator.

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

Step 1.26

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

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Step 1.26.1

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

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

Step 1.26.2

Combine the numerators over the common denominator.

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

Step 1.26.3

Add $1$ and $1$ .

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

Step 1.26.4

Divide $2$ by $2$ .

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

Step 1.27

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

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

Step 1.28

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

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

Step 1.29

Reorder terms.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 2.3

Simplify.

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

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $- 2 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [4]$ .

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Multiply $2$ by $- 2$ .

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

Step 2.4.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.4.6

Add $- 4 x$ and $0$ .

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

Step 2.5

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

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Step 2.5.1

To apply the Chain Rule, set $u$ as $- x^{2} + 4$ .

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

Step 2.5.2

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

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

Step 2.5.3

Replace all occurrences of $u$ with $- x^{2} + 4$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Combine fractions.

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Step 2.10.1

Move the negative in front of the fraction.

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

Step 2.10.2

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

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

Step 2.10.3

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

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

Step 2.11

By the Sum Rule, the derivative of $- x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4]$ .

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Multiply $2$ by $- 1$ .

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

Step 2.15

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.16

Simplify terms.

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Step 2.16.1

Add $- 2 x$ and $0$ .

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

Step 2.16.2

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

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

Step 2.16.3

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

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

Step 2.16.4

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

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

Step 2.17

Cancel the common factors.

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Step 2.17.1

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

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

Step 2.17.2

Cancel the common factor.

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

Step 2.17.3

Rewrite the expression.

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

Step 2.18

Move the negative in front of the fraction.

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

Step 2.19

Multiply $- 1$ by $- 1$ .

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

Step 2.20

Multiply $- 2 x^{2} + 4$ by $1$ .

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

Step 2.21

Simplify.

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Step 2.21.1

Simplify the numerator.

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Step 2.21.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.21.1.2

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

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

Step 2.21.1.3

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

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Step 2.21.1.3.1

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

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

Step 2.21.1.3.2

Factor $2$ out of $4$ .

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

Step 2.21.1.3.3

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

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

Step 2.21.1.4

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

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

Step 2.21.1.5

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

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

Step 2.21.1.6

Combine the numerators over the common denominator.

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

Step 2.21.1.7

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

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Step 2.21.1.7.1

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

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Step 2.21.1.7.1.1

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

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

Step 2.21.1.7.1.2

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

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

Step 2.21.1.7.1.3

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

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

Step 2.21.1.7.2

Combine exponents.

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Step 2.21.1.7.2.1

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

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Step 2.21.1.7.2.1.1

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

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

Step 2.21.1.7.2.1.2

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

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

Step 2.21.1.7.2.1.3

Combine the numerators over the common denominator.

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

Step 2.21.1.7.2.1.4

Add $1$ and $1$ .

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

Step 2.21.1.7.2.1.5

Divide $2$ by $2$ .

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

Step 2.21.1.7.2.2

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

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

Step 2.21.1.8

Simplify the numerator.

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Step 2.21.1.8.1

Apply the distributive property.

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

Step 2.21.1.8.2

Multiply $- 1$ by $- 2$ .

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

Step 2.21.1.8.3

Multiply $- 2$ by $4$ .

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

Step 2.21.1.8.4

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

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

Step 2.21.1.8.5

Add $- 8$ and $2$ .

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

Step 2.21.2

Combine terms.

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Step 2.21.2.1

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

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

Step 2.21.2.2

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

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

Step 2.21.2.3

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

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Step 2.21.2.3.1

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

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Step 2.21.2.3.1.1

Raise $- x^{2} + 4$ to the power of $1$ .

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

Step 2.21.2.3.1.2

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

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

Step 2.21.2.3.2

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

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

Step 2.21.2.3.3

Combine the numerators over the common denominator.

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

Step 2.21.2.3.4

Add $1$ and $2$ .

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

Step 3

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

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

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

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

Step 4.1.2

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

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 $g (x) = 4 - x^{2}$ .

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Step 4.1.3.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

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

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 $n = \frac{1}{2}$ .

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

Step 4.1.3.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

Combine the numerators over the common denominator.

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

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

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

Step 4.1.7.2

Subtract $2$ from $1$ .

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

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

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

Step 4.1.8.2

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

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

Step 4.1.8.3

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

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

Step 4.1.8.4

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

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

Step 4.1.9

By the Sum Rule, the derivative of $4 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

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

Step 4.1.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 4.1.11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 4.1.12

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

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

Step 4.1.13

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

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

Step 4.1.14

Combine fractions.

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Step 4.1.14.1

Multiply $2$ by $- 1$ .

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

Step 4.1.14.2

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

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

Step 4.1.14.3

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

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

Step 4.1.15

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

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

Step 4.1.16

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

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

Step 4.1.17

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

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

Step 4.1.18

Add $1$ and $1$ .

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

Step 4.1.19

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

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

Step 4.1.20

Cancel the common factors.

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Step 4.1.20.1

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

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

Step 4.1.20.2

Cancel the common factor.

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

Step 4.1.20.3

Rewrite the expression.

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

Step 4.1.21

Move the negative in front of the fraction.

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

Step 4.1.22

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

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

Step 4.1.23

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

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

Step 4.1.24

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

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

Step 4.1.25

Combine the numerators over the common denominator.

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

Step 4.1.26

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

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Step 4.1.26.1

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

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

Step 4.1.26.2

Combine the numerators over the common denominator.

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

Step 4.1.26.3

Add $1$ and $1$ .

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

Step 4.1.26.4

Divide $2$ by $2$ .

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

Step 4.1.27

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

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

Step 4.1.28

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

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

Step 4.1.29

Reorder terms.

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

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

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Subtract $4$ from both sides of the equation.

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

Step 5.3.2

Divide each term in $- 2 x^{2} = - 4$ by $- 2$ and simplify.

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Step 5.3.2.1

Divide each term in $- 2 x^{2} = - 4$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 4}{- 2}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 4}{- 2}$

Step 5.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 2}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 4$ by $- 2$ .

$x^{2} = 2$

Step 5.3.3

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

$x = \pm \sqrt{2}$

Step 5.3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 5.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 5.3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

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.

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

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{- 2 x^{2} + 4}{\sqrt{- x^{2} + 4}}$

Step 6.2

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

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

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.

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

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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Step 6.3.2.2.1.1.1

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

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

Step 6.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.2

Simplify.

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

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 $0$ .

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

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Step 6.3.3.2

Divide each term in $- x^{2} = - 4$ by $- 1$ and simplify.

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Step 6.3.3.2.1

Divide each term in $- x^{2} = - 4$ by $- 1$ .

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

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.3.3.2.2.2

Divide $x^{2}$ by $1$ .

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

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Step 6.3.3.3

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

$x = \pm \sqrt{4}$

Step 6.3.3.4

Simplify $\pm \sqrt{4}$ .

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Step 6.3.3.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 6.3.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 6.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 6.3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 6.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 6.4

Set the radicand in $\sqrt{- x^{2} + 4}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 4 < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Subtract $4$ from both sides of the inequality.

$- x^{2} < - 4$

Step 6.5.2

Divide each term in $- x^{2} < - 4$ by $- 1$ and simplify.

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Step 6.5.2.1

Divide each term in $- x^{2} < - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 4}{- 1}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 4}{- 1}$

Step 6.5.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{- 4}{- 1}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} > 4$

Step 6.5.3

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

$\sqrt{x^{2}} > \sqrt{4}$

Step 6.5.4

Simplify the equation.

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Step 6.5.4.1

Simplify the left side.

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Step 6.5.4.1.1

Pull terms out from under the radical.

$| x | > \sqrt{4}$

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

Simplify $\sqrt{4}$ .

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Step 6.5.4.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | > \sqrt{2^{2}}$

Step 6.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 2 |$

Step 6.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | > 2$

Step 6.5.5

Write $| x | > 2$ as a piecewise.

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Step 6.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 2$

Step 6.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 2$

Step 6.5.5.5

Write as a piecewise.

${\begin{cases} x > 2 & x \geq 0 \\ - x > 2 & x < 0 \end{cases}$

Step 6.5.6

Find the intersection of $x > 2$ and $x \geq 0$ .

$x > 2$

Step 6.5.7

Divide each term in $- x > 2$ by $- 1$ and simplify.

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Step 6.5.7.1

Divide each term in $- x > 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{2}{- 1}$

Step 6.5.7.2

Simplify the left side.

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Step 6.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{2}{- 1}$

Step 6.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{2}{- 1}$

Step 6.5.7.3

Simplify the right side.

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Step 6.5.7.3.1

Divide $2$ by $- 1$ .

$x < - 2$

Step 6.5.8

Find the union of the solutions.

$x < - 2$ or $x > 2$

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 $0$ .

$x \leq - 2, x \geq 2$

$(- \infty, - 2] \cup [2, \infty)$

$x \leq - 2, x \geq 2$

$(- \infty, - 2] \cup [2, \infty)$

Step 7

Critical points to evaluate.

$x = \sqrt{2}, - \sqrt{2}, - 2, 2$

Step 8

Evaluate the second derivative at $x = \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.1.1.1

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

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

Step 9.1.1.2

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

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

Step 9.1.1.3

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

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

Step 9.1.1.4

Cancel the common factor of $2$ .

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Step 9.1.1.4.1

Cancel the common factor.

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

Step 9.1.1.4.2

Rewrite the expression.

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

Step 9.1.1.5

Evaluate the exponent.

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

Step 9.1.2

Subtract $6$ from $2$ .

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

Step 9.1.3

Multiply $- 4$ by $2$ .

$\frac{- 8 \sqrt{2}}{{(- {\sqrt{2}}^{2} + 4)}^{\frac{3}{2}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.2.1.1.1

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

$\frac{- 8 \sqrt{2}}{{(- {(2^{\frac{1}{2}})}^{2} + 4)}^{\frac{3}{2}}}$

Step 9.2.1.1.2

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

$\frac{- 8 \sqrt{2}}{{(- 2^{\frac{1}{2} \cdot 2} + 4)}^{\frac{3}{2}}}$

Step 9.2.1.1.3

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

$\frac{- 8 \sqrt{2}}{{(- 2^{\frac{2}{2}} + 4)}^{\frac{3}{2}}}$

Step 9.2.1.1.4

Cancel the common factor of $2$ .

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Step 9.2.1.1.4.1

Cancel the common factor.

$\frac{- 8 \sqrt{2}}{{(- 2^{\frac{2}{2}} + 4)}^{\frac{3}{2}}}$

Step 9.2.1.1.4.2

Rewrite the expression.

$\frac{- 8 \sqrt{2}}{{(- 2^{1} + 4)}^{\frac{3}{2}}}$

Step 9.2.1.1.5

Evaluate the exponent.

$\frac{- 8 \sqrt{2}}{{(- 1 \cdot 2 + 4)}^{\frac{3}{2}}}$

Step 9.2.1.2

Multiply $- 1$ by $2$ .

$\frac{- 8 \sqrt{2}}{{(- 2 + 4)}^{\frac{3}{2}}}$

Step 9.2.2

Add $- 2$ and $4$ .

$\frac{- 8 \sqrt{2}}{2^{\frac{3}{2}}}$

Step 9.3

Move the negative in front of the fraction.

$- \frac{8 \sqrt{2}}{2^{\frac{3}{2}}}$

Step 10

$x = \sqrt{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \sqrt{2}$ is a local maximum

Step 11

Find the y-value when $x = \sqrt{2}$ .

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Step 11.1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

Combine using the product rule for radicals.

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

Step 11.2.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 11.2.2.1

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

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

Step 11.2.2.2

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

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

Step 11.2.2.3

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

$f (\sqrt{2}) = \sqrt{2 (4 - 2^{\frac{2}{2}})}$

Step 11.2.2.4

Cancel the common factor of $2$ .

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Step 11.2.2.4.1

Cancel the common factor.

$f (\sqrt{2}) = \sqrt{2 (4 - 2^{\frac{2}{2}})}$

Step 11.2.2.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2 (4 - 2)}$

Step 11.2.2.5

Evaluate the exponent.

$f (\sqrt{2}) = \sqrt{2 (4 - 1 \cdot 2)}$

Step 11.2.3

Simplify the expression.

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Step 11.2.3.1

Multiply $- 1$ by $2$ .

$f (\sqrt{2}) = \sqrt{2 (4 - 2)}$

Step 11.2.3.2

Subtract $2$ from $4$ .

$f (\sqrt{2}) = \sqrt{2 \cdot 2}$

Step 11.2.3.3

Multiply $2$ by $2$ .

$f (\sqrt{2}) = \sqrt{4}$

Step 11.2.3.4

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{2}) = \sqrt{2^{2}}$

Step 11.2.4

Pull terms out from under the radical, assuming positive real numbers.

$f (\sqrt{2}) = 2$

Step 11.2.5

The final answer is $2$ .

$y = 2$

Step 12

Evaluate the second derivative at $x = - \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $- 1$ by $2$ .

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

Step 13.2

Simplify the denominator.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Apply the product rule to $- \sqrt{2}$ .

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

Step 13.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 13.2.1.2.1

Move ${(- 1)}^{2}$ .

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

Step 13.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 13.2.1.2.2.1

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

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

Step 13.2.1.2.2.2

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

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

Step 13.2.1.2.3

Add $2$ and $1$ .

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

Step 13.2.1.3

Raise $- 1$ to the power of $3$ .

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

Step 13.2.1.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 13.2.1.4.1

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

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

Step 13.2.1.4.2

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

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

Step 13.2.1.4.3

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

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

Step 13.2.1.4.4

Cancel the common factor of $2$ .

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Step 13.2.1.4.4.1

Cancel the common factor.

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

Step 13.2.1.4.4.2

Rewrite the expression.

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

Step 13.2.1.4.5

Evaluate the exponent.

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

Step 13.2.1.5

Multiply $- 1$ by $2$ .

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

Step 13.2.2

Add $- 2$ and $4$ .

$\frac{- 2 \sqrt{2} ({(- \sqrt{2})}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Apply the product rule to $- \sqrt{2}$ .

$\frac{- 2 \sqrt{2} ({(- 1)}^{2} {\sqrt{2}}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.2

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

$\frac{- 2 \sqrt{2} (1 {\sqrt{2}}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$\frac{- 2 \sqrt{2} ({\sqrt{2}}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 13.3.4.1

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

$\frac{- 2 \sqrt{2} ({(2^{\frac{1}{2}})}^{2} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.4.2

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

$\frac{- 2 \sqrt{2} (2^{\frac{1}{2} \cdot 2} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.4.3

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

$\frac{- 2 \sqrt{2} (2^{\frac{2}{2}} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.4.4

Cancel the common factor of $2$ .

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Step 13.3.4.4.1

Cancel the common factor.

$\frac{- 2 \sqrt{2} (2^{\frac{2}{2}} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.4.4.2

Rewrite the expression.

$\frac{- 2 \sqrt{2} (2^{1} - 6)}{2^{\frac{3}{2}}}$

Step 13.3.4.5

Evaluate the exponent.

$\frac{- 2 \sqrt{2} (2 - 6)}{2^{\frac{3}{2}}}$

Step 13.3.5

Subtract $6$ from $2$ .

$\frac{- 2 \sqrt{2} \cdot - 4}{2^{\frac{3}{2}}}$

Step 13.3.6

Multiply $- 4$ by $- 2$ .

$\frac{8 \sqrt{2}}{2^{\frac{3}{2}}}$

Step 14

$x = - \sqrt{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \sqrt{2}$ is a local minimum

Step 15

Find the y-value when $x = - \sqrt{2}$ .

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Step 15.1

Replace the variable $x$ with $- \sqrt{2}$ in the expression.

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

Step 15.2

Simplify the result.

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Step 15.2.1

Apply the product rule to $- \sqrt{2}$ .

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

Step 15.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 15.2.2.1

Move ${(- 1)}^{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 + {(- 1)}^{2} \cdot (- 1 {\sqrt{2}}^{2})}$

Step 15.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 15.2.2.2.1

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{2}}^{2})}$

Step 15.2.2.2.2

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

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

Step 15.2.2.3

Add $2$ and $1$ .

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

Step 15.2.3

Raise $- 1$ to the power of $3$ .

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

Step 15.2.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.2.4.1

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

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

Step 15.2.4.2

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

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

Step 15.2.4.3

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2^{\frac{2}{2}}}$

Step 15.2.4.4

Cancel the common factor of $2$ .

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Step 15.2.4.4.1

Cancel the common factor.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2^{\frac{2}{2}}}$

Step 15.2.4.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2}$

Step 15.2.4.5

Evaluate the exponent.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 1 \cdot 2}$

Step 15.2.5

Simplify the expression.

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Step 15.2.5.1

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{4 - 2}$

Step 15.2.5.2

Subtract $2$ from $4$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2}$

Step 15.2.6

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 15.2.6.1

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

$f (- \sqrt{2}) = - (\sqrt{2} \sqrt{2})$

Step 15.2.6.2

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

$f (- \sqrt{2}) = - (\sqrt{2} \sqrt{2})$

Step 15.2.6.3

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

$f (- \sqrt{2}) = - {\sqrt{2}}^{1 + 1}$

Step 15.2.6.4

Add $1$ and $1$ .

$f (- \sqrt{2}) = - {\sqrt{2}}^{2}$

Step 15.2.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.2.7.1

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

$f (- \sqrt{2}) = - {(2^{\frac{1}{2}})}^{2}$

Step 15.2.7.2

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

$f (- \sqrt{2}) = - 2^{\frac{1}{2} \cdot 2}$

Step 15.2.7.3

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

$f (- \sqrt{2}) = - 2^{\frac{2}{2}}$

Step 15.2.7.4

Cancel the common factor of $2$ .

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Step 15.2.7.4.1

Cancel the common factor.

$f (- \sqrt{2}) = - 2^{\frac{2}{2}}$

Step 15.2.7.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - 2$

Step 15.2.7.5

Evaluate the exponent.

$f (- \sqrt{2}) = - 1 \cdot 2$

Step 15.2.8

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = - 2$

Step 15.2.9

The final answer is $- 2$ .

$y = - 2$

Step 16

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

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

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

Step 17.1.2

Multiply $- 1$ by $4$ .

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

Step 17.2

Reduce the expression by cancelling the common factors.

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Step 17.2.1

Add $- 4$ and $4$ .

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{\frac{3}{2}}}$

Step 17.2.2

Simplify the expression.

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Step 17.2.2.1

Rewrite $0$ as $0^{2}$ .

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{{(0^{2})}^{\frac{3}{2}}}$

Step 17.2.2.2

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

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{2 (\frac{3}{2})}}$

Step 17.2.3

Cancel the common factor of $2$ .

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Step 17.2.3.1

Cancel the common factor.

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{2 (\frac{3}{2})}}$

Step 17.2.3.2

Rewrite the expression.

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0^{3}}$

Step 17.2.4

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

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0}$

Step 17.2.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{2 (- 2) ({(- 2)}^{2} - 6)}{0}$

Step 17.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 18

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 19