Calculus Examples

$f (x) = 2 x - \sqrt{16 - x^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x - \sqrt{16 - x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 1.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \sqrt{16 - x^{2}}]$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 - x^{2}}$ as ${(16 - x^{2})}^{\frac{1}{2}}$ .

$2 + \frac{d}{d x} [- {(16 - x^{2})}^{\frac{1}{2}}]$

Step 1.3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(16 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(16 - x^{2})}^{\frac{1}{2}}]$ .

$2 - \frac{d}{d x} [{(16 - x^{2})}^{\frac{1}{2}}]$

Step 1.3.3

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 $g (x) = 16 - x^{2}$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u$ as $16 - x^{2}$ .

$2 - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [16 - x^{2}])$

Step 1.3.3.2

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

$2 - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [16 - x^{2}])$

Step 1.3.3.3

Replace all occurrences of $u$ with $16 - x^{2}$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [16 - x^{2}])$

Step 1.3.4

By the Sum Rule, the derivative of $16 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]))$

Step 1.3.5

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [- x^{2}]))$

Step 1.3.6

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (0 - \frac{d}{d x} [x^{2}]))$

Step 1.3.7

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (0 - (2 x)))$

Step 1.3.8

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - (2 x)))$

Step 1.3.9

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - (2 x)))$

Step 1.3.10

Combine the numerators over the common denominator.

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 - (2 x)))$

Step 1.3.11

Simplify the numerator.

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Step 1.3.11.1

Multiply $- 1$ by $2$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1 - 2}{2}} (0 - (2 x)))$

Step 1.3.11.2

Subtract $2$ from $1$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{- 1}{2}} (0 - (2 x)))$

Step 1.3.12

Move the negative in front of the fraction.

$2 - (\frac{1}{2} {(16 - x^{2})}^{- \frac{1}{2}} (0 - (2 x)))$

Step 1.3.13

Multiply $2$ by $- 1$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{- \frac{1}{2}} (0 - 2 x))$

Step 1.3.14

Subtract $2 x$ from $0$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{- \frac{1}{2}} (- 2 x))$

Step 1.3.15

Combine $\frac{1}{2}$ and ${(16 - x^{2})}^{- \frac{1}{2}}$ .

$2 - (\frac{{(16 - x^{2})}^{- \frac{1}{2}}}{2} (- 2 x))$

Step 1.3.16

Combine $- 2$ and $\frac{{(16 - x^{2})}^{- \frac{1}{2}}}{2}$ .

$2 - (\frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}}}{2} x)$

Step 1.3.17

Combine $\frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}}}{2}$ and $x$ .

$2 - \frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}} x}{2}$

Step 1.3.18

Move ${(16 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 - \frac{- 2 x}{2 {(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.3.19

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

$2 - \frac{2 (- x)}{2 {(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.3.20

Cancel the common factors.

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Step 1.3.20.1

Factor $2$ out of $2 {(16 - x^{2})}^{\frac{1}{2}}$ .

$2 - \frac{2 (- x)}{2 ({(16 - x^{2})}^{\frac{1}{2}})}$

Step 1.3.20.2

Cancel the common factor.

$2 - \frac{2 (- x)}{2 {(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.3.20.3

Rewrite the expression.

$2 - \frac{- x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.3.21

Move the negative in front of the fraction.

$2 - - \frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.3.22

Multiply $- 1$ by $- 1$ .

$2 + 1 \frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.3.23

Multiply $\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$ by $1$ .

$2 + \frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 1.4

Reorder terms.

$\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2$ with respect to $x$ is $\frac{d}{d x} [\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}] + \frac{d}{d x} [2]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}]$ .

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Step 2.2.1

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 $f (x) = x$ and $g (x) = {(16 - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.2.2

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

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

Step 2.2.3

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 $g (x) = 16 - x^{2}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $16 - x^{2}$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u$ with $16 - x^{2}$ .

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

Step 2.2.4

By the Sum Rule, the derivative of $16 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]$ .

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

Step 2.2.5

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} \cdot 1 - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} (- x^{2}))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.6

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

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} \cdot 1 - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1}{2} - 1} (0 - \frac{d}{d x} x^{2})))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.7

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

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} \cdot 1 - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1}{2} - 1} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.8

Multiply ${(16 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1}{2} - 1} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.9

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

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.10

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

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.11

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.12

Simplify the numerator.

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Step 2.2.12.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{1 - 2}{2}} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.12.2

Subtract $2$ from $1$ .

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{\frac{- 1}{2}} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.13

Move the negative in front of the fraction.

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{- \frac{1}{2}} (0 - (2 x))))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.14

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(16 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot ({(16 - x^{2})}^{- \frac{1}{2}} (0 - 2 x)))}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.15

Subtract $2 x$ from $0$ .

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

Step 2.2.16

Combine $\frac{1}{2}$ and ${(16 - x^{2})}^{- \frac{1}{2}}$ .

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

Step 2.2.17

Combine $- 2$ and $\frac{{(16 - x^{2})}^{- \frac{1}{2}}}{2}$ .

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

Step 2.2.18

Combine $\frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}}}{2}$ and $x$ .

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

Step 2.2.19

Move ${(16 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2.20

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

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

Step 2.2.21

Cancel the common factors.

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Step 2.2.21.1

Factor $2$ out of $2 {(16 - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.2.21.2

Cancel the common factor.

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

Step 2.2.21.3

Rewrite the expression.

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

Step 2.2.22

Move the negative in front of the fraction.

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

Step 2.2.23

Multiply $- 1$ by $- 1$ .

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

Step 2.2.24

Multiply $x$ by $1$ .

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

Step 2.2.25

Combine $x$ and $\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 2.2.26

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

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

Step 2.2.27

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

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

Step 2.2.28

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

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

Step 2.2.29

Add $1$ and $1$ .

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

Step 2.2.30

To write ${(16 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(16 - x^{2})}^{\frac{1}{2}}}{{(16 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 2.2.31

Combine the numerators over the common denominator.

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

Step 2.2.32

Multiply ${(16 - x^{2})}^{\frac{1}{2}}$ by ${(16 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.32.1

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

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

Step 2.2.32.2

Combine the numerators over the common denominator.

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

Step 2.2.32.3

Add $1$ and $1$ .

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

Step 2.2.32.4

Divide $2$ by $2$ .

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

Step 2.2.33

Simplify ${(16 - x^{2})}^{1}$ .

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

Step 2.2.34

Add $- x^{2}$ and $x^{2}$ .

$f'' (x) = \frac{\frac{16 + 0}{{(16 - x^{2})}^{\frac{1}{2}}}}{{({(16 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{d}{d x} (2)$

Step 2.2.35

Add $16$ and $0$ .

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

Step 2.2.36

Multiply the exponents in ${({(16 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.36.1

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

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

Step 2.2.36.2

Cancel the common factor of $2$ .

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Step 2.2.36.2.1

Cancel the common factor.

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

Step 2.2.36.2.2

Rewrite the expression.

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

Step 2.2.37

Simplify.

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

Step 2.2.38

Rewrite $\frac{\frac{16}{{(16 - x^{2})}^{\frac{1}{2}}}}{16 - x^{2}}$ as a product.

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

Step 2.2.39

Multiply $\frac{16}{{(16 - x^{2})}^{\frac{1}{2}}}$ by $\frac{1}{16 - x^{2}}$ .

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

Step 2.2.40

Multiply ${(16 - x^{2})}^{\frac{1}{2}}$ by $16 - x^{2}$ by adding the exponents.

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Step 2.2.40.1

Multiply ${(16 - x^{2})}^{\frac{1}{2}}$ by $16 - x^{2}$ .

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Step 2.2.40.1.1

Raise $16 - x^{2}$ to the power of $1$ .

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

Step 2.2.40.1.2

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

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

Step 2.2.40.2

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

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

Step 2.2.40.3

Combine the numerators over the common denominator.

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

Step 2.2.40.4

Add $1$ and $2$ .

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

Step 2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = \frac{16}{{(16 - x^{2})}^{\frac{3}{2}}} + 0$

Step 2.4

Simplify.

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Step 2.4.1

Add $\frac{16}{{(16 - x^{2})}^{\frac{3}{2}}}$ and $0$ .

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

Step 2.4.2

Reorder terms.

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

Step 3

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

$\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2 = 0$

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

By the Sum Rule, the derivative of $2 x - \sqrt{16 - x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.1.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 4.1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 4.1.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- \sqrt{16 - x^{2}}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- \sqrt{16 - x^{2}}]$ .

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Step 4.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 - x^{2}}$ as ${(16 - x^{2})}^{\frac{1}{2}}$ .

$2 + \frac{d}{d x} [- {(16 - x^{2})}^{\frac{1}{2}}]$

Step 4.1.3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(16 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(16 - x^{2})}^{\frac{1}{2}}]$ .

$2 - \frac{d}{d x} [{(16 - x^{2})}^{\frac{1}{2}}]$

Step 4.1.3.3

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 $g (x) = 16 - x^{2}$ .

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Step 4.1.3.3.1

To apply the Chain Rule, set $u$ as $16 - x^{2}$ .

$2 - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [16 - x^{2}])$

Step 4.1.3.3.2

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

$2 - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [16 - x^{2}])$

Step 4.1.3.3.3

Replace all occurrences of $u$ with $16 - x^{2}$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [16 - x^{2}])$

Step 4.1.3.4

By the Sum Rule, the derivative of $16 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]))$

Step 4.1.3.5

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [- x^{2}]))$

Step 4.1.3.6

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (0 - \frac{d}{d x} [x^{2}]))$

Step 4.1.3.7

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1} (0 - (2 x)))$

Step 4.1.3.8

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - (2 x)))$

Step 4.1.3.9

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

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - (2 x)))$

Step 4.1.3.10

Combine the numerators over the common denominator.

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 - (2 x)))$

Step 4.1.3.11

Simplify the numerator.

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Step 4.1.3.11.1

Multiply $- 1$ by $2$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{1 - 2}{2}} (0 - (2 x)))$

Step 4.1.3.11.2

Subtract $2$ from $1$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{\frac{- 1}{2}} (0 - (2 x)))$

Step 4.1.3.12

Move the negative in front of the fraction.

$2 - (\frac{1}{2} {(16 - x^{2})}^{- \frac{1}{2}} (0 - (2 x)))$

Step 4.1.3.13

Multiply $2$ by $- 1$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{- \frac{1}{2}} (0 - 2 x))$

Step 4.1.3.14

Subtract $2 x$ from $0$ .

$2 - (\frac{1}{2} {(16 - x^{2})}^{- \frac{1}{2}} (- 2 x))$

Step 4.1.3.15

Combine $\frac{1}{2}$ and ${(16 - x^{2})}^{- \frac{1}{2}}$ .

$2 - (\frac{{(16 - x^{2})}^{- \frac{1}{2}}}{2} (- 2 x))$

Step 4.1.3.16

Combine $- 2$ and $\frac{{(16 - x^{2})}^{- \frac{1}{2}}}{2}$ .

$2 - (\frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}}}{2} x)$

Step 4.1.3.17

Combine $\frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}}}{2}$ and $x$ .

$2 - \frac{- 2 {(16 - x^{2})}^{- \frac{1}{2}} x}{2}$

Step 4.1.3.18

Move ${(16 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 - \frac{- 2 x}{2 {(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.3.19

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

$2 - \frac{2 (- x)}{2 {(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.3.20

Cancel the common factors.

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Step 4.1.3.20.1

Factor $2$ out of $2 {(16 - x^{2})}^{\frac{1}{2}}$ .

$2 - \frac{2 (- x)}{2 ({(16 - x^{2})}^{\frac{1}{2}})}$

Step 4.1.3.20.2

Cancel the common factor.

$2 - \frac{2 (- x)}{2 {(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.3.20.3

Rewrite the expression.

$2 - \frac{- x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.3.21

Move the negative in front of the fraction.

$2 - - \frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.3.22

Multiply $- 1$ by $- 1$ .

$2 + 1 \frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.3.23

Multiply $\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$ by $1$ .

$2 + \frac{x}{{(16 - x^{2})}^{\frac{1}{2}}}$

Step 4.1.4

Reorder terms.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2$ .

$\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x}{{(16 - x^{2})}^{\frac{1}{2}}} + 2 = 0$

Step 5.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 3.57770876$

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.

$\frac{x}{\sqrt{{(16 - x^{2})}^{1}}} + 2$

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{x}{\sqrt{16 - x^{2}}} + 2$

Step 6.2

Set the denominator in $\frac{x}{\sqrt{16 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{16 - x^{2}} = 0$

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.

${\sqrt{16 - x^{2}}}^{2} = 0^{2}$

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 $\sqrt{16 - x^{2}}$ as ${(16 - x^{2})}^{\frac{1}{2}}$ .

${({(16 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${({(16 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${({(16 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1.1

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

${(16 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

${(16 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

${(16 - x^{2})}^{1} = 0^{2}$

Step 6.3.2.2.1.2

Simplify.

$16 - x^{2} = 0^{2}$

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 $0$ .

$16 - x^{2} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Subtract $16$ from both sides of the equation.

$- x^{2} = - 16$

Step 6.3.3.2

Divide each term in $- x^{2} = - 16$ by $- 1$ and simplify.

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Step 6.3.3.2.1

Divide each term in $- x^{2} = - 16$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 16}{- 1}$

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.

$\frac{x^{2}}{1} = \frac{- 16}{- 1}$

Step 6.3.3.2.2.2

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

$x^{2} = \frac{- 16}{- 1}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Divide $- 16$ by $- 1$ .

$x^{2} = 16$

Step 6.3.3.3

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

$x = \pm \sqrt{16}$

Step 6.3.3.4

Simplify $\pm \sqrt{16}$ .

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Step 6.3.3.4.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 6.3.3.4.2

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

$x = \pm 4$

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.

$x = 4$

Step 6.3.3.5.2

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

$x = - 4$

Step 6.3.3.5.3

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

$x = 4, - 4$

Step 6.4

Set the radicand in $\sqrt{16 - x^{2}}$ less than $0$ to find where the expression is undefined.

$16 - x^{2} < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Subtract $16$ from both sides of the inequality.

$- x^{2} < - 16$

Step 6.5.2

Divide each term in $- x^{2} < - 16$ by $- 1$ and simplify.

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Step 6.5.2.1

Divide each term in $- x^{2} < - 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 16}{- 1}$

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.

$\frac{x^{2}}{1} > \frac{- 16}{- 1}$

Step 6.5.2.2.2

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

$x^{2} > \frac{- 16}{- 1}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Divide $- 16$ by $- 1$ .

$x^{2} > 16$

Step 6.5.3

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

$\sqrt{x^{2}} > \sqrt{16}$

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.

$| x | > \sqrt{16}$

Step 6.5.4.2

Simplify the right side.

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Step 6.5.4.2.1

Simplify $\sqrt{16}$ .

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Step 6.5.4.2.1.1

Rewrite $16$ as $4^{2}$ .

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

Step 6.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 4 |$

Step 6.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$| x | > 4$

Step 6.5.5

Write $| x | > 4$ 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.

$x > 4$

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 $- 1$ .

$- x > 4$

Step 6.5.5.5

Write as a piecewise.

${\begin{cases} x > 4 & x \geq 0 \\ - x > 4 & x < 0 \end{cases}$

Step 6.5.6

Find the intersection of $x > 4$ and $x \geq 0$ .

$x > 4$

Step 6.5.7

Divide each term in $- x > 4$ by $- 1$ and simplify.

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Step 6.5.7.1

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

$\frac{- x}{- 1} < \frac{4}{- 1}$

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.

$\frac{x}{1} < \frac{4}{- 1}$

Step 6.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{4}{- 1}$

Step 6.5.7.3

Simplify the right side.

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Step 6.5.7.3.1

Divide $4$ by $- 1$ .

$x < - 4$

Step 6.5.8

Find the union of the solutions.

$x < - 4$ or $x > 4$

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 $0$ .

$x \leq - 4, x \geq 4$

$(- \infty, - 4] \cup [4, \infty)$

$x \leq - 4, x \geq 4$

$(- \infty, - 4] \cup [4, \infty)$

Step 7

Critical points to evaluate.

$x = - 3.57770876, - 4, 4$

Step 8

Evaluate the second derivative at $x = - 3.57770876$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{16}{{(- {(- 3.57770876)}^{2} + 16)}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Simplify each term.

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Step 9.1.1.1

Raise $- 3.57770876$ to the power of $2$ .

$\frac{16}{{(- 1 \cdot 12.8 + 16)}^{\frac{3}{2}}}$

Step 9.1.1.2

Multiply $- 1$ by $12.8$ .

$\frac{16}{{(- 12.8 + 16)}^{\frac{3}{2}}}$

Step 9.1.2

Add $- 12.8$ and $16$ .

$\frac{16}{{3.19999999}^{\frac{3}{2}}}$

Step 9.1.3

Rewrite $3.19999999$ as ${1.78885438}^{2}$ .

$\frac{16}{{({1.78885438}^{2})}^{\frac{3}{2}}}$

Step 9.1.4

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

$\frac{16}{{1.78885438}^{2 (\frac{3}{2})}}$

Step 9.1.5

Cancel the common factor of $2$ .

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Step 9.1.5.1

Cancel the common factor.

$\frac{16}{{1.78885438}^{2 (\frac{3}{2})}}$

Step 9.1.5.2

Rewrite the expression.

$\frac{16}{{1.78885438}^{3}}$

Step 9.1.6

Raise $1.78885438$ to the power of $3$ .

$\frac{16}{5.72433402}$

Step 9.2

Divide $16$ by $5.72433402$ .

$2.79508497$

Step 10

$x = - 3.57770876$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 3.57770876$ is a local minimum

Step 11

Find the y-value when $x = - 3.57770876$ .

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Step 11.1

Replace the variable $x$ with $- 3.57770876$ in the expression.

$f (- 3.57770876) = 2 (- 3.57770876) - \sqrt{16 - {(- 3.57770876)}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $2$ by $- 3.57770876$ .

$f (- 3.57770876) = - 7.15541752 - \sqrt{16 - {(- 3.57770876)}^{2}}$

Step 11.2.1.2

Raise $- 3.57770876$ to the power of $2$ .

$f (- 3.57770876) = - 7.15541752 - \sqrt{16 - 1 \cdot 12.8}$

Step 11.2.1.3

Multiply $- 1$ by $12.8$ .

$f (- 3.57770876) = - 7.15541752 - \sqrt{16 - 12.8}$

Step 11.2.1.4

Subtract $12.8$ from $16$ .

$f (- 3.57770876) = - 7.15541752 - \sqrt{3.19999999}$

Step 11.2.2

Subtract $\sqrt{3.19999999}$ from $- 7.15541752$ .

$f (- 3.57770876) = - 8.9442719$

Step 11.2.3

The final answer is $- 8.9442719$ .

$y = - 8.9442719$

Step 12

Evaluate the second derivative at $x = - 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{16}{{(- {(- 4)}^{2} + 16)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Raise $- 4$ to the power of $2$ .

$\frac{16}{{(- 1 \cdot 16 + 16)}^{\frac{3}{2}}}$

Step 13.1.2

Multiply $- 1$ by $16$ .

$\frac{16}{{(- 16 + 16)}^{\frac{3}{2}}}$

Step 13.2

Reduce the expression by cancelling the common factors.

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Step 13.2.1

Add $- 16$ and $16$ .

$\frac{16}{0^{\frac{3}{2}}}$

Step 13.2.2

Simplify the expression.

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Step 13.2.2.1

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

$\frac{16}{{(0^{2})}^{\frac{3}{2}}}$

Step 13.2.2.2

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

$\frac{16}{0^{2 (\frac{3}{2})}}$

Step 13.2.3

Cancel the common factor of $2$ .

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Step 13.2.3.1

Cancel the common factor.

$\frac{16}{0^{2 (\frac{3}{2})}}$

Step 13.2.3.2

Rewrite the expression.

$\frac{16}{0^{3}}$

Step 13.2.4

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

$\frac{16}{0}$

Step 13.2.5

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

Undefined

$\frac{16}{0}$

Step 13.3

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