Calculus Examples

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

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

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

Step 1.2

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 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Simplify terms.

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Step 1.13.1

Multiply $2$ by $- 1$ .

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

Step 1.13.2

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

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

Step 1.13.3

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

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

Step 1.13.4

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

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

Step 1.14

Cancel the common factors.

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Step 1.14.1

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

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

Step 1.14.2

Cancel the common factor.

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

Step 1.14.3

Rewrite the expression.

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

Step 1.15

Move the negative in front of the fraction.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

Differentiate using the Power Rule.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.6

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 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.10.2

Subtract $2$ from $1$ .

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

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

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

Step 2.11.2

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

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

Step 2.11.3

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

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

Step 2.11.4

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Multiply.

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Step 2.16.1

Multiply $- 1$ by $- 1$ .

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

Step 2.16.2

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

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

Step 2.17

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

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

Step 2.18

Combine fractions.

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Step 2.18.1

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

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

Step 2.18.2

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

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

Step 2.19

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

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

Step 2.20

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

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

Step 2.21

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

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

Step 2.22

Add $1$ and $1$ .

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

Step 2.23

Cancel the common factor.

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

Step 2.24

Rewrite the expression.

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

Step 2.25

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

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

Step 2.26

Combine the numerators over the common denominator.

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

Step 2.27

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

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Step 2.27.1

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

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

Step 2.27.2

Combine the numerators over the common denominator.

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

Step 2.27.3

Add $1$ and $1$ .

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

Step 2.27.4

Divide $2$ by $2$ .

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

Step 2.28

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

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

Step 2.29

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

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

Step 2.30

Add $4$ and $0$ .

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

Step 2.31

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

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

Step 2.32

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

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

Step 2.33

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

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Step 2.33.1

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

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Step 2.33.1.1

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

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

Step 2.33.1.2

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

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

Step 2.33.2

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

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

Step 2.33.3

Combine the numerators over the common denominator.

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

Step 2.33.4

Add $1$ and $2$ .

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

Step 2.34

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

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

Step 2.35

Simplify the expression.

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Step 2.35.1

Multiply $\frac{x}{{(4 - x^{2})}^{\frac{1}{2}}}$ by $0$ .

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

Step 2.35.2

Add $- \frac{4}{{(4 - x^{2})}^{\frac{3}{2}}}$ and $0$ .

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

Step 3

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

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

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

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

Step 4.1.2

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 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $- 1$ by $2$ .

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

Step 4.1.6.2

Subtract $2$ from $1$ .

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

Step 4.1.7

Combine fractions.

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Step 4.1.7.1

Move the negative in front of the fraction.

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

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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

Step 4.1.8

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

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

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

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

Step 4.1.12

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

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

Step 4.1.13

Simplify terms.

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Step 4.1.13.1

Multiply $2$ by $- 1$ .

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

Step 4.1.13.2

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

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

Step 4.1.13.3

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

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

Step 4.1.13.4

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

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

Step 4.1.14

Cancel the common factors.

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Step 4.1.14.1

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

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

Step 4.1.14.2

Cancel the common factor.

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

Step 4.1.14.3

Rewrite the expression.

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

Step 4.1.15

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$x = 0$

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.

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

Step 6.1.2

Anything raised to $1$ is the base itself.

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

Step 6.2

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

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

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.

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

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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Step 6.3.2.2.1.1.1

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

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

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.

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

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.2

Simplify.

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

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 $0$ .

$4 - x^{2} = 0$

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

$4 - x^{2} < 0$

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.

$x = 0, - 2, 2$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

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

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

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

Step 9.1.1.2

Multiply $- 1$ by $0$ .

$- \frac{4}{{(4 + 0)}^{\frac{3}{2}}}$

Step 9.1.2

Add $4$ and $0$ .

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

Step 9.1.3

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

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

Step 9.1.4

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

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

Step 9.1.5

Cancel the common factor of $2$ .

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Step 9.1.5.1

Cancel the common factor.

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

Step 9.1.5.2

Rewrite the expression.

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

Step 9.1.6

Raise $2$ to the power of $3$ .

$- \frac{4}{8}$

Step 9.2

Cancel the common factor of $4$ and $8$ .

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Step 9.2.1

Factor $4$ out of $4$ .

$- \frac{4 (1)}{8}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $4$ out of $8$ .

$- \frac{4 \cdot 1}{4 \cdot 2}$

Step 9.2.2.2

Cancel the common factor.

$- \frac{4 \cdot 1}{4 \cdot 2}$

Step 9.2.2.3

Rewrite the expression.

$- \frac{1}{2}$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

Replace the variable $x$ with $0$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

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

$f (0) = \sqrt{4 - 0}$

Step 11.2.2

Multiply $- 1$ by $0$ .

$f (0) = \sqrt{4 + 0}$

Step 11.2.3

Add $4$ and $0$ .

$f (0) = \sqrt{4}$

Step 11.2.4

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

$f (0) = \sqrt{2^{2}}$

Step 11.2.5

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

$f (0) = 2$

Step 11.2.6

The final answer is $2$ .

$y = 2$

Step 12

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.

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

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

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

Step 13.1.2

Multiply $- 1$ by $4$ .

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

Step 13.2

Reduce the expression by cancelling the common factors.

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Step 13.2.1

Subtract $4$ from $4$ .

$- \frac{4}{0^{\frac{3}{2}}}$

Step 13.2.2

Simplify the expression.

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Step 13.2.2.1

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

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

Step 13.2.2.2

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

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

Step 13.2.3

Cancel the common factor of $2$ .

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Step 13.2.3.1

Cancel the common factor.

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

Step 13.2.3.2

Rewrite the expression.

$- \frac{4}{0^{3}}$

Step 13.2.4

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

$- \frac{4}{0}$

Step 13.2.5

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

Undefined

$- \frac{4}{0}$

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