Calculus Examples

$\sqrt{9 - x^{2}}$

Step 1

Write $\sqrt{9 - x^{2}}$ as a function.

$f (x) = \sqrt{9 - x^{2}}$

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(9 - x^{2})}^{\frac{1}{2}}]$

Step 2.2

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 $g (x) = 9 - x^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $9 - x^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [9 - x^{2}]$

Step 2.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x^{2}]$

Step 2.2.3

Replace all occurrences of $u$ with $9 - x^{2}$ .

$\frac{1}{2} {(9 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x^{2}]$

Step 2.3

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

$\frac{1}{2} {(9 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 2.4

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

$\frac{1}{2} {(9 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(9 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(9 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(9 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(9 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 2.7.2

Combine $\frac{1}{2}$ and ${(9 - x^{2})}^{- \frac{1}{2}}$ .

$\frac{{(9 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [9 - x^{2}]$

Step 2.7.3

Move ${(9 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [9 - x^{2}]$

Step 2.8

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}])$

Step 2.9

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 2.10

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

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}]$

Step 2.11

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

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}])$

Step 2.12

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

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (- (2 x))$

Step 2.13

Simplify terms.

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Step 2.13.1

Multiply $2$ by $- 1$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (- 2 x)$

Step 2.13.2

Combine $- 2$ and $\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{- 2}{2 {(9 - x^{2})}^{\frac{1}{2}}} x$

Step 2.13.3

Combine $\frac{- 2}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{- 2 x}{2 {(9 - x^{2})}^{\frac{1}{2}}}$

Step 2.13.4

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

$\frac{2 (- x)}{2 {(9 - x^{2})}^{\frac{1}{2}}}$

Step 2.14

Cancel the common factors.

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Step 2.14.1

Factor $2$ out of $2 {(9 - x^{2})}^{\frac{1}{2}}$ .

$\frac{2 (- x)}{2 ({(9 - x^{2})}^{\frac{1}{2}})}$

Step 2.14.2

Cancel the common factor.

$\frac{2 (- x)}{2 {(9 - x^{2})}^{\frac{1}{2}}}$

Step 2.14.3

Rewrite the expression.

$\frac{- x}{{(9 - x^{2})}^{\frac{1}{2}}}$

Step 2.15

Move the negative in front of the fraction.

$- \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

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 $g (x) = \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{d}{d x} \cdot \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.2

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 $g (x) = {(9 - x^{2})}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(9 - x^{2})}^{\frac{1}{2}}}{{({(9 - x^{2})}^{\frac{1}{2}})}^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.3

Multiply the exponents in ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(9 - x^{2})}^{\frac{1}{2}}}{{(9 - x^{2})}^{\frac{1}{2} \cdot 2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

Step 3.3.2.2

Rewrite the expression.

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(9 - x^{2})}^{\frac{1}{2}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.4

Simplify.

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} (x) - x \frac{d}{d x} {(9 - x^{2})}^{\frac{1}{2}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5

Differentiate using the Power Rule.

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Step 3.5.1

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} \cdot 1 - x \frac{d}{d x} {(9 - x^{2})}^{\frac{1}{2}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.5.2

Multiply ${(9 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x \frac{d}{d x} {(9 - x^{2})}^{\frac{1}{2}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.6

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 $g (x) = 9 - x^{2}$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $9 - x^{2}$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.6.2

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.6.3

Replace all occurrences of $u$ with $9 - x^{2}$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.7

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.8

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.9

Combine the numerators over the common denominator.

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.10.2

Subtract $2$ from $1$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11

Combine fractions.

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Step 3.11.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} \cdot {(9 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11.2

Combine $\frac{1}{2}$ and ${(9 - x^{2})}^{- \frac{1}{2}}$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{{(9 - x^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11.3

Move ${(9 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - x (\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.11.4

Combine $\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x^{2})}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.12

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (9) + \frac{d}{d x} (- x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.13

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x^{2}))}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.14

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2})}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.15

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2})}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.16

Multiply.

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Step 3.16.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + 1 (\frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}}) \cdot \frac{d}{d x} (x^{2})}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.16.2

Multiply $\frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ by $1$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x^{2})}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot (2 x)}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.18

Combine fractions.

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Step 3.18.1

Combine $2$ and $\frac{x}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 x}{2 {(9 - x^{2})}^{\frac{1}{2}}} \cdot x}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.18.2

Combine $\frac{2 x}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 x \cdot x}{2 {(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.19

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 (x \cdot x)}{2 {(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.20

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 (x \cdot x)}{2 {(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.21

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

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 x^{1 + 1}}{2 {(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.22

Add $1$ and $1$ .

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 x^{2}}{2 {(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.23

Cancel the common factor.

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{2 x^{2}}{2 {(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.24

Rewrite the expression.

$f'' (x) = - \frac{{(9 - x^{2})}^{\frac{1}{2}} + \frac{x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.25

To write ${(9 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(9 - x^{2})}^{\frac{1}{2}}}{{(9 - x^{2})}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{\frac{{(9 - x^{2})}^{\frac{1}{2}} {(9 - x^{2})}^{\frac{1}{2}}}{{(9 - x^{2})}^{\frac{1}{2}}} + \frac{x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.26

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{{(9 - x^{2})}^{\frac{1}{2}} {(9 - x^{2})}^{\frac{1}{2}} + x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.27

Multiply ${(9 - x^{2})}^{\frac{1}{2}}$ by ${(9 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 3.27.1

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

$f'' (x) = - \frac{\frac{{(9 - x^{2})}^{\frac{1}{2} + \frac{1}{2}} + x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.27.2

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{{(9 - x^{2})}^{\frac{1 + 1}{2}} + x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.27.3

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{{(9 - x^{2})}^{\frac{2}{2}} + x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.27.4

Divide $2$ by $2$ .

$f'' (x) = - \frac{\frac{(9 - x^{2}) + x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.28

Simplify ${(9 - x^{2})}^{1}$ .

$f'' (x) = - \frac{\frac{9 - x^{2} + x^{2}}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.29

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

$f'' (x) = - \frac{\frac{9 + 0}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.30

Add $9$ and $0$ .

$f'' (x) = - \frac{\frac{9}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.31

Rewrite $\frac{\frac{9}{{(9 - x^{2})}^{\frac{1}{2}}}}{9 - x^{2}}$ as a product.

$f'' (x) = - (\frac{9}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{1}{9 - x^{2}}) + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.32

Multiply $\frac{9}{{(9 - x^{2})}^{\frac{1}{2}}}$ by $\frac{1}{9 - x^{2}}$ .

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{1}{2}} (9 - x^{2})} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.33

Multiply ${(9 - x^{2})}^{\frac{1}{2}}$ by $9 - x^{2}$ by adding the exponents.

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Step 3.33.1

Multiply ${(9 - x^{2})}^{\frac{1}{2}}$ by $9 - x^{2}$ .

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Step 3.33.1.1

Raise $9 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{1}{2}} (9 - x^{2})} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.33.1.2

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

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{1}{2} + 1}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.33.2

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

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{1}{2} + \frac{2}{2}}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.33.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{1 + 2}{2}}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.33.4

Add $1$ and $2$ .

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{3}{2}}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.34

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

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{3}{2}}} + \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} \cdot 0$

Step 3.35

Simplify the expression.

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Step 3.35.1

Multiply $\frac{x}{{(9 - x^{2})}^{\frac{1}{2}}}$ by $0$ .

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{3}{2}}} + 0$

Step 3.35.2

Add $- \frac{9}{{(9 - x^{2})}^{\frac{3}{2}}}$ and $0$ .

$f'' (x) = - \frac{9}{{(9 - x^{2})}^{\frac{3}{2}}}$

Step 4

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

$- \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(9 - x^{2})}^{\frac{1}{2}}]$

Step 5.1.2

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 $g (x) = 9 - x^{2}$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $9 - x^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [9 - x^{2}]$

Step 5.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.2.3

Replace all occurrences of $u$ with $9 - x^{2}$ .

$\frac{1}{2} {(9 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.3

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

$\frac{1}{2} {(9 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.4

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

$\frac{1}{2} {(9 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(9 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(9 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(9 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.7

Combine fractions.

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Step 5.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(9 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.7.2

Combine $\frac{1}{2}$ and ${(9 - x^{2})}^{- \frac{1}{2}}$ .

$\frac{{(9 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.7.3

Move ${(9 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [9 - x^{2}]$

Step 5.1.8

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}])$

Step 5.1.9

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 5.1.10

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

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}]$

Step 5.1.11

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

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}])$

Step 5.1.12

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

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (- (2 x))$

Step 5.1.13

Simplify terms.

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Step 5.1.13.1

Multiply $2$ by $- 1$ .

$\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}} (- 2 x)$

Step 5.1.13.2

Combine $- 2$ and $\frac{1}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{- 2}{2 {(9 - x^{2})}^{\frac{1}{2}}} x$

Step 5.1.13.3

Combine $\frac{- 2}{2 {(9 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{- 2 x}{2 {(9 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.13.4

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

$\frac{2 (- x)}{2 {(9 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.14

Cancel the common factors.

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Step 5.1.14.1

Factor $2$ out of $2 {(9 - x^{2})}^{\frac{1}{2}}$ .

$\frac{2 (- x)}{2 ({(9 - x^{2})}^{\frac{1}{2}})}$

Step 5.1.14.2

Cancel the common factor.

$\frac{2 (- x)}{2 {(9 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.14.3

Rewrite the expression.

$\frac{- x}{{(9 - x^{2})}^{\frac{1}{2}}}$

Step 5.1.15

Move the negative in front of the fraction.

$f' (x) = - \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}}$ .

$- \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{x}{{(9 - x^{2})}^{\frac{1}{2}}} = 0$

Step 6.2

Set the numerator equal to zero.

$x = 0$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

$- \frac{x}{\sqrt{{(9 - x^{2})}^{1}}}$

Step 7.1.2

Anything raised to $1$ is the base itself.

$- \frac{x}{\sqrt{9 - x^{2}}}$

Step 7.2

Set the denominator in $\frac{x}{\sqrt{9 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{9 - x^{2}} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

${\sqrt{9 - x^{2}}}^{2} = 0^{2}$

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

${({(9 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Simplify ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 7.3.2.2.1.1

Multiply the exponents in ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 7.3.2.2.1.1.1

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

${(9 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 7.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1.1.2.1

Cancel the common factor.

${(9 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 7.3.2.2.1.1.2.2

Rewrite the expression.

${(9 - x^{2})}^{1} = 0^{2}$

Step 7.3.2.2.1.2

Simplify.

$9 - x^{2} = 0^{2}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$9 - x^{2} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Subtract $9$ from both sides of the equation.

$- x^{2} = - 9$

Step 7.3.3.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 7.3.3.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 7.3.3.2.2

Simplify the left side.

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Step 7.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 7.3.3.2.2.2

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

$x^{2} = \frac{- 9}{- 1}$

Step 7.3.3.2.3

Simplify the right side.

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Step 7.3.3.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 7.3.3.3

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

$x = \pm \sqrt{9}$

Step 7.3.3.4

Simplify $\pm \sqrt{9}$ .

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Step 7.3.3.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 7.3.3.4.2

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

$x = \pm 3$

Step 7.3.3.5

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

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Step 7.3.3.5.1

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

$x = 3$

Step 7.3.3.5.2

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

$x = - 3$

Step 7.3.3.5.3

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

$x = 3, - 3$

Step 7.4

Set the radicand in $\sqrt{9 - x^{2}}$ less than $0$ to find where the expression is undefined.

$9 - x^{2} < 0$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Subtract $9$ from both sides of the inequality.

$- x^{2} < - 9$

Step 7.5.2

Divide each term in $- x^{2} < - 9$ by $- 1$ and simplify.

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Step 7.5.2.1

Divide each term in $- x^{2} < - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 9}{- 1}$

Step 7.5.2.2

Simplify the left side.

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Step 7.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 9}{- 1}$

Step 7.5.2.2.2

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

$x^{2} > \frac{- 9}{- 1}$

Step 7.5.2.3

Simplify the right side.

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Step 7.5.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} > 9$

Step 7.5.3

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

$\sqrt{x^{2}} > \sqrt{9}$

Step 7.5.4

Simplify the equation.

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Step 7.5.4.1

Simplify the left side.

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Step 7.5.4.1.1

Pull terms out from under the radical.

$| x | > \sqrt{9}$

Step 7.5.4.2

Simplify the right side.

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Step 7.5.4.2.1

Simplify $\sqrt{9}$ .

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Step 7.5.4.2.1.1

Rewrite $9$ as $3^{2}$ .

$| x | > \sqrt{3^{2}}$

Step 7.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 3 |$

Step 7.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | > 3$

Step 7.5.5

Write $| x | > 3$ as a piecewise.

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Step 7.5.5.1

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

$x \geq 0$

Step 7.5.5.2

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

$x > 3$

Step 7.5.5.3

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

$x < 0$

Step 7.5.5.4

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

$- x > 3$

Step 7.5.5.5

Write as a piecewise.

${\begin{cases} x > 3 & x \geq 0 \\ - x > 3 & x < 0 \end{cases}$

Step 7.5.6

Find the intersection of $x > 3$ and $x \geq 0$ .

$x > 3$

Step 7.5.7

Divide each term in $- x > 3$ by $- 1$ and simplify.

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Step 7.5.7.1

Divide each term in $- x > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{3}{- 1}$

Step 7.5.7.2

Simplify the left side.

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Step 7.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{3}{- 1}$

Step 7.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{3}{- 1}$

Step 7.5.7.3

Simplify the right side.

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Step 7.5.7.3.1

Divide $3$ by $- 1$ .

$x < - 3$

Step 7.5.8

Find the union of the solutions.

$x < - 3$ or $x > 3$

Step 7.6

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 $0$ .

$x \leq - 3, x \geq 3$

$(- \infty, - 3] \cup [3, \infty)$

$x \leq - 3, x \geq 3$

$(- \infty, - 3] \cup [3, \infty)$

Step 8

Critical points to evaluate.

$x = 0, - 3, 3$

Step 9

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.

$- \frac{9}{{(9 - {(0)}^{2})}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

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

$- \frac{9}{{(9 - 0)}^{\frac{3}{2}}}$

Step 10.1.1.2

Multiply $- 1$ by $0$ .

$- \frac{9}{{(9 + 0)}^{\frac{3}{2}}}$

Step 10.1.2

Add $9$ and $0$ .

$- \frac{9}{9^{\frac{3}{2}}}$

Step 10.1.3

Rewrite $9$ as $3^{2}$ .

$- \frac{9}{{(3^{2})}^{\frac{3}{2}}}$

Step 10.1.4

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

$- \frac{9}{3^{2 (\frac{3}{2})}}$

Step 10.1.5

Cancel the common factor of $2$ .

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Step 10.1.5.1

Cancel the common factor.

$- \frac{9}{3^{2 (\frac{3}{2})}}$

Step 10.1.5.2

Rewrite the expression.

$- \frac{9}{3^{3}}$

Step 10.1.6

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

$- \frac{9}{27}$

Step 10.2

Cancel the common factor of $9$ and $27$ .

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Step 10.2.1

Factor $9$ out of $9$ .

$- \frac{9 (1)}{27}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $9$ out of $27$ .

$- \frac{9 \cdot 1}{9 \cdot 3}$

Step 10.2.2.2

Cancel the common factor.

$- \frac{9 \cdot 1}{9 \cdot 3}$

Step 10.2.2.3

Rewrite the expression.

$- \frac{1}{3}$

Step 11

$x = 0$ 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 12

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

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Step 12.1

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

$f (0) = \sqrt{9 - {(0)}^{2}}$

Step 12.2

Simplify the result.

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Step 12.2.1

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

$f (0) = \sqrt{9 - 0}$

Step 12.2.2

Multiply $- 1$ by $0$ .

$f (0) = \sqrt{9 + 0}$

Step 12.2.3

Add $9$ and $0$ .

$f (0) = \sqrt{9}$

Step 12.2.4

Rewrite $9$ as $3^{2}$ .

$f (0) = \sqrt{3^{2}}$

Step 12.2.5

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

$f (0) = 3$

Step 12.2.6

The final answer is $3$ .

$y = 3$

Step 13

Evaluate the second derivative at $x = - 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{9}{{(9 - {(- 3)}^{2})}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

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

$- \frac{9}{{(9 - 1 \cdot 9)}^{\frac{3}{2}}}$

Step 14.1.2

Multiply $- 1$ by $9$ .

$- \frac{9}{{(9 - 9)}^{\frac{3}{2}}}$

Step 14.2

Reduce the expression by cancelling the common factors.

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Step 14.2.1

Subtract $9$ from $9$ .

$- \frac{9}{0^{\frac{3}{2}}}$

Step 14.2.2

Simplify the expression.

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Step 14.2.2.1

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

$- \frac{9}{{(0^{2})}^{\frac{3}{2}}}$

Step 14.2.2.2

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

$- \frac{9}{0^{2 (\frac{3}{2})}}$

Step 14.2.3

Cancel the common factor of $2$ .

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Step 14.2.3.1

Cancel the common factor.

$- \frac{9}{0^{2 (\frac{3}{2})}}$

Step 14.2.3.2

Rewrite the expression.

$- \frac{9}{0^{3}}$

Step 14.2.4

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

$- \frac{9}{0}$

Step 14.2.5

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

Undefined

$- \frac{9}{0}$

Step 14.3

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

Undefined

Step 15

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

No Local Extrema

Step 16