Calculus Examples

$f (x) = | 7 - x^{2} |$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 7 - x^{2}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $7 - x^{2}$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [7 - x^{2}]$

Step 1.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [7 - x^{2}]$

Step 1.1.3

Replace all occurrences of $u$ with $7 - x^{2}$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} \frac{d}{d x} [7 - x^{2}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $7 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- x^{2}]$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} (\frac{d}{d x} [7] + \frac{d}{d x} [- x^{2}])$

Step 1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} (0 + \frac{d}{d x} [- x^{2}])$

Step 1.2.3

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

$\frac{7 - x^{2}}{| 7 - x^{2} |} \frac{d}{d x} [- x^{2}]$

Step 1.2.4

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

$\frac{7 - x^{2}}{| 7 - x^{2} |} (- \frac{d}{d x} [x^{2}])$

Step 1.2.5

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

$\frac{7 - x^{2}}{| 7 - x^{2} |} (- (2 x))$

Step 1.2.6

Combine fractions.

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Step 1.2.6.1

Multiply $2$ by $- 1$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} (- 2 x)$

Step 1.2.6.2

Combine $- 2$ and $\frac{7 - x^{2}}{| 7 - x^{2} |}$ .

$\frac{- 2 (7 - x^{2})}{| 7 - x^{2} |} x$

Step 1.2.6.3

Combine $\frac{- 2 (7 - x^{2})}{| 7 - x^{2} |}$ and $x$ .

$\frac{- 2 (7 - x^{2}) x}{| 7 - x^{2} |}$

Step 1.2.6.4

Move the negative in front of the fraction.

$- \frac{(2 (7 - x^{2})) x}{| 7 - x^{2} |}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$- \frac{(2 \cdot 7 + 2 (- x^{2})) x}{| 7 - x^{2} |}$

Step 1.3.2

Apply the distributive property.

$- \frac{2 \cdot 7 x + 2 (- x^{2}) x}{| 7 - x^{2} |}$

Step 1.3.3

Simplify each term.

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Step 1.3.3.1

Multiply $2$ by $7$ .

$- \frac{14 x + 2 (- x^{2}) x}{| 7 - x^{2} |}$

Step 1.3.3.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.3.2.1

Move $x$ .

$- \frac{14 x + 2 (- (x \cdot x^{2}))}{| 7 - x^{2} |}$

Step 1.3.3.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.3.2.2.1

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

$- \frac{14 x + 2 (- (x^{1} x^{2}))}{| 7 - x^{2} |}$

Step 1.3.3.2.2.2

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

$- \frac{14 x + 2 (- x^{1 + 2})}{| 7 - x^{2} |}$

Step 1.3.3.2.3

Add $1$ and $2$ .

$- \frac{14 x + 2 (- x^{3})}{| 7 - x^{2} |}$

Step 1.3.3.3

Multiply $- 1$ by $2$ .

$- \frac{14 x - 2 x^{3}}{| 7 - x^{2} |}$

Step 1.3.4

Factor $2 x$ out of $14 x - 2 x^{3}$ .

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Step 1.3.4.1

Factor $2 x$ out of $14 x$ .

$- \frac{2 x (7) - 2 x^{3}}{| 7 - x^{2} |}$

Step 1.3.4.2

Factor $2 x$ out of $- 2 x^{3}$ .

$- \frac{2 x (7) + 2 x (- x^{2})}{| 7 - x^{2} |}$

Step 1.3.4.3

Factor $2 x$ out of $2 x (7) + 2 x (- x^{2})$ .

$- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x (7 - x^{2})}{| 7 - x^{2} |}]$ .

$f'' (x) = - 2 \frac{d}{d x} \frac{x (7 - x^{2})}{| 7 - x^{2} |}$

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 $f (x) = x (7 - x^{2})$ and $g (x) = | 7 - x^{2} |$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | \frac{d}{d x} (x (7 - x^{2})) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 7 - x^{2}$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x \frac{d}{d x} (7 - x^{2}) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $7 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- x^{2}]$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x (\frac{d}{d x} (7) + \frac{d}{d x} (- x^{2})) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.4.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x (0 + \frac{d}{d x} (- x^{2})) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.4.3

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x \frac{d}{d x} (- x^{2}) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.4.4

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x (- \frac{d}{d x} x^{2}) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.4.5

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x (- (2 x)) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.4.6

Multiply $2$ by $- 1$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (x (- 2 x) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.5

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 2 (x \cdot x) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.6

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 2 (x \cdot x) + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.7

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 2 x^{1 + 1} + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.8

Differentiate using the Power Rule.

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Step 2.8.1

Add $1$ and $1$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 2 x^{2} + (7 - x^{2}) \frac{d}{d x} (x)) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.8.2

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 2 x^{2} + (7 - x^{2}) \cdot 1) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.8.3

Simplify by adding terms.

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Step 2.8.3.1

Multiply $7 - x^{2}$ by $1$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 2 x^{2} + 7 - x^{2}) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.8.3.2

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - x (7 - x^{2}) \frac{d}{d x} | 7 - x^{2} |}{{| 7 - x^{2} |}^{2}}$

Step 2.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 7 - x^{2}$ .

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Step 2.9.1

To apply the Chain Rule, set $u$ as $7 - x^{2}$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - x (7 - x^{2}) (\frac{d}{d u} (| u |) \frac{d}{d x} (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.9.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - x (7 - x^{2}) (\frac{u}{| u |} \cdot \frac{d}{d x} (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.9.3

Replace all occurrences of $u$ with $7 - x^{2}$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - x (7 - x^{2}) (\frac{7 - x^{2}}{| 7 - x^{2} |} \cdot \frac{d}{d x} (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.10

Differentiate.

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Step 2.10.1

Combine $\frac{7 - x^{2}}{| 7 - x^{2} |}$ and $x$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot (7 - x^{2}) \frac{d}{d x} (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.10.2

By the Sum Rule, the derivative of $7 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- x^{2}]$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot ((7 - x^{2}) (\frac{d}{d x} (7) + \frac{d}{d x} (- x^{2})))}{{| 7 - x^{2} |}^{2}}$

Step 2.10.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot ((7 - x^{2}) (0 + \frac{d}{d x} (- x^{2})))}{{| 7 - x^{2} |}^{2}}$

Step 2.10.4

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot (7 - x^{2}) \frac{d}{d x} (- x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.10.5

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) - \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot ((7 - x^{2}) (- \frac{d}{d x} x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.10.6

Multiply.

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Step 2.10.6.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + 1 (\frac{(7 - x^{2}) x}{| 7 - x^{2} |}) \cdot (7 - x^{2}) \frac{d}{d x} (x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.10.6.2

Multiply $\frac{(7 - x^{2}) x}{| 7 - x^{2} |}$ by $1$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot (7 - x^{2}) \frac{d}{d x} (x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.10.7

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{(7 - x^{2}) x}{| 7 - x^{2} |} \cdot ((7 - x^{2}) (2 x))}{{| 7 - x^{2} |}^{2}}$

Step 2.10.8

Combine fractions.

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Step 2.10.8.1

Combine $2$ and $\frac{(7 - x^{2}) x}{| 7 - x^{2} |}$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{2 ((7 - x^{2}) x)}{| 7 - x^{2} |} \cdot ((7 - x^{2}) x)}{{| 7 - x^{2} |}^{2}}$

Step 2.10.8.2

Combine $x$ and $\frac{2 ((7 - x^{2}) x)}{| 7 - x^{2} |}$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x (2 ((7 - x^{2}) x))}{| 7 - x^{2} |} \cdot (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.11

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x \cdot x (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.12

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x \cdot x (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.13

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

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{1 + 1} (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.14

Add $1$ and $1$ .

$f'' (x) = - 2 \frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{2} (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$

Step 2.15

Combine $- 2$ and $\frac{| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{2} (2 (7 - x^{2}))}{| 7 - x^{2} |} (7 - x^{2})}{{| 7 - x^{2} |}^{2}}$ .

$f'' (x) = \frac{- 2 (| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{2} (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.16

Move the negative in front of the fraction.

$f'' (x) = - \frac{2 (| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{2} (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17

Simplify.

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Step 2.17.1

Apply the distributive property.

$f'' (x) = - \frac{2 (| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{2} (2 \cdot 7 + 2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.2

Apply the distributive property.

$f'' (x) = - \frac{2 (| 7 - x^{2} | (- 3 x^{2} + 7) + \frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.3

Apply the distributive property.

$f'' (x) = - \frac{2 (| 7 - x^{2} | (- 3 x^{2} + 7)) + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4

Simplify the numerator.

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Step 2.17.4.1

Simplify each term.

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Step 2.17.4.1.1

Apply the distributive property.

$f'' (x) = - \frac{2 (| 7 - x^{2} | (- 3 x^{2}) + | 7 - x^{2} | \cdot 7) + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{2 (- 3 | 7 - x^{2} | x^{2} + | 7 - x^{2} | \cdot 7) + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.3

Move $7$ to the left of $| 7 - x^{2} |$ .

$f'' (x) = - \frac{2 (- 3 | 7 - x^{2} | x^{2} + 7 \cdot | 7 - x^{2} |) + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.4

Apply the distributive property.

$f'' (x) = - \frac{2 (- 3 | 7 - x^{2} | x^{2}) + 2 (7 | 7 - x^{2} |) + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.5

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{- 6 (| 7 - x^{2} | x^{2}) + 2 (7 | 7 - x^{2} |) + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.6

Multiply $7$ by $2$ .

$f'' (x) = - \frac{- 6 (| 7 - x^{2} | x^{2}) + 14 | 7 - x^{2} | + 2 (\frac{x^{2} (2 \cdot 7) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.7

Factor $x^{2} \cdot 2$ out of $x^{2} \cdot 2 \cdot 7 + x^{2} (2 (- x^{2}))$ .

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Step 2.17.4.1.7.1

Factor $x^{2} \cdot 2$ out of $x^{2} \cdot 2 \cdot 7$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{x^{2} \cdot (2 (7)) + x^{2} (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.7.2

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

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{x^{2} \cdot (2 (7)) + x^{2} \cdot (2 (- x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.7.3

Factor $x^{2} \cdot 2$ out of $x^{2} \cdot 2 (7) + x^{2} \cdot 2 (- x^{2})$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{x^{2} \cdot (2 (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.8

Move $2$ to the left of $x^{2}$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{2 \cdot (x^{2} (7 - x^{2}))}{| 7 - x^{2} |} \cdot (7 - x^{2}))}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.9

Multiply $\frac{2 x^{2} (7 - x^{2})}{| 7 - x^{2} |}$ by $7 - x^{2}$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{2 x^{2} (7 - x^{2}) (7 - x^{2})}{| 7 - x^{2} |})}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.10

Simplify the numerator.

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Step 2.17.4.1.10.1

Raise $7 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{2 x^{2} ((7 - x^{2}) (7 - x^{2}))}{| 7 - x^{2} |})}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.10.2

Raise $7 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{2 x^{2} ((7 - x^{2}) (7 - x^{2}))}{| 7 - x^{2} |})}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.10.3

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

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{2 x^{2} {(7 - x^{2})}^{1 + 1}}{| 7 - x^{2} |})}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.10.4

Add $1$ and $1$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + 2 (\frac{2 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |})}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.11

Multiply $2 \frac{2 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}$ .

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Step 2.17.4.1.11.1

Combine $2$ and $\frac{2 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + \frac{2 (2 x^{2} {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.1.11.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + \frac{4 (x^{2} {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

$f'' (x) = - \frac{- 6 | 7 - x^{2} | x^{2} + 14 | 7 - x^{2} | + \frac{4 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.2

To write $- 6 | 7 - x^{2} | x^{2}$ as a fraction with a common denominator, multiply by $\frac{| 7 - x^{2} |}{| 7 - x^{2} |}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | - 6 | 7 - x^{2} | x^{2} \cdot \frac{| 7 - x^{2} |}{| 7 - x^{2} |} + \frac{4 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.3

Combine $- 6 | 7 - x^{2} | x^{2}$ and $\frac{| 7 - x^{2} |}{| 7 - x^{2} |}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{- 6 | 7 - x^{2} | x^{2} | 7 - x^{2} |}{| 7 - x^{2} |} + \frac{4 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{- 6 | 7 - x^{2} | x^{2} | 7 - x^{2} | + 4 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5

Simplify the numerator.

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Step 2.17.4.5.1

Factor $2 x^{2}$ out of $- 6 | 7 - x^{2} | x^{2} | 7 - x^{2} | + 4 x^{2} {(7 - x^{2})}^{2}$ .

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Step 2.17.4.5.1.1

Factor $2 x^{2}$ out of $- 6 | 7 - x^{2} | x^{2} | 7 - x^{2} |$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 7 - x^{2} | | 7 - x^{2} |) + 4 x^{2} {(7 - x^{2})}^{2}}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.1.2

Factor $2 x^{2}$ out of $4 x^{2} {(7 - x^{2})}^{2}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 7 - x^{2} | | 7 - x^{2} |) + 2 x^{2} (2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.1.3

Factor $2 x^{2}$ out of $2 x^{2} (- 3 | 7 - x^{2} | | 7 - x^{2} |) + 2 x^{2} (2 {(7 - x^{2})}^{2})$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 7 - x^{2} | | 7 - x^{2} | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.2

Combine exponents.

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Step 2.17.4.5.2.1

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | (7 - x^{2}) (7 - x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.2.2

Raise $7 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | (7 - x^{2}) (7 - x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.2.3

Raise $7 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | (7 - x^{2}) (7 - x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.2.4

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

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | {(7 - x^{2})}^{1 + 1} | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.2.5

Add $1$ and $1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | {(7 - x^{2})}^{2} | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3

Simplify each term.

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Step 2.17.4.5.3.1

Rewrite ${(7 - x^{2})}^{2}$ as $(7 - x^{2}) (7 - x^{2})$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | (7 - x^{2}) (7 - x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.2

Expand $(7 - x^{2}) (7 - x^{2})$ using the FOIL Method.

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Step 2.17.4.5.3.2.1

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 7 (7 - x^{2}) - x^{2} (7 - x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.2.2

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 7 \cdot 7 + 7 (- x^{2}) - x^{2} (7 - x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.2.3

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 7 \cdot 7 + 7 (- x^{2}) - x^{2} \cdot 7 - x^{2} (- x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3

Simplify and combine like terms.

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Step 2.17.4.5.3.3.1

Simplify each term.

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Step 2.17.4.5.3.3.1.1

Multiply $7$ by $7$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 + 7 (- x^{2}) - x^{2} \cdot 7 - x^{2} (- x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.2

Multiply $- 1$ by $7$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - x^{2} \cdot 7 - x^{2} (- x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.3

Multiply $7$ by $- 1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} - x^{2} (- x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{2} x^{2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.17.4.5.3.3.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 (x^{2} x^{2})) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.5.2

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

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{2 + 2}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{4}) | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} + 1 x^{4} | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.1.7

Multiply $x^{4}$ by $1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 7 x^{2} - 7 x^{2} + x^{4} | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.3.2

Subtract $7 x^{2}$ from $- 7 x^{2}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 {(7 - x^{2})}^{2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.4

Rewrite ${(7 - x^{2})}^{2}$ as $(7 - x^{2}) (7 - x^{2})$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 ((7 - x^{2}) (7 - x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.5

Expand $(7 - x^{2}) (7 - x^{2})$ using the FOIL Method.

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Step 2.17.4.5.3.5.1

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (7 (7 - x^{2}) - x^{2} (7 - x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.5.2

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (7 \cdot 7 + 7 (- x^{2}) - x^{2} (7 - x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.5.3

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (7 \cdot 7 + 7 (- x^{2}) - x^{2} \cdot 7 - x^{2} (- x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6

Simplify and combine like terms.

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Step 2.17.4.5.3.6.1

Simplify each term.

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Step 2.17.4.5.3.6.1.1

Multiply $7$ by $7$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 + 7 (- x^{2}) - x^{2} \cdot 7 - x^{2} (- x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.2

Multiply $- 1$ by $7$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - x^{2} \cdot 7 - x^{2} (- x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.3

Multiply $7$ by $- 1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} - x^{2} (- x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{2} x^{2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.17.4.5.3.6.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 (x^{2} x^{2}))))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.5.2

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

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{2 + 2})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{4})))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} + 1 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.1.7

Multiply $x^{4}$ by $1$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 7 x^{2} - 7 x^{2} + x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.6.2

Subtract $7 x^{2}$ from $- 7 x^{2}$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 (49 - 14 x^{2} + x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.7

Apply the distributive property.

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 2 \cdot 49 + 2 (- 14 x^{2}) + 2 x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.8

Simplify.

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Step 2.17.4.5.3.8.1

Multiply $2$ by $49$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 + 2 (- 14 x^{2}) + 2 x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.5.3.8.2

Multiply $- 14$ by $2$ .

$f'' (x) = - \frac{14 | 7 - x^{2} | + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.6

To write $14 | 7 - x^{2} |$ as a fraction with a common denominator, multiply by $\frac{| 7 - x^{2} |}{| 7 - x^{2} |}$ .

$f'' (x) = - \frac{\frac{14 | 7 - x^{2} | | 7 - x^{2} |}{| 7 - x^{2} |} + \frac{2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.7

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{14 | 7 - x^{2} | | 7 - x^{2} | + 2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8

Simplify the numerator.

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Step 2.17.4.8.1

Factor $2$ out of $14 | 7 - x^{2} | | 7 - x^{2} | + 2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4})$ .

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Step 2.17.4.8.1.1

Factor $2$ out of $14 | 7 - x^{2} | | 7 - x^{2} |$ .

$f'' (x) = - \frac{\frac{2 (7 | 7 - x^{2} | | 7 - x^{2} |) + 2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.1.2

Factor $2$ out of $2 x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4})$ .

$f'' (x) = - \frac{\frac{2 (7 | 7 - x^{2} | | 7 - x^{2} |) + 2 (x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.1.3

Factor $2$ out of $2 (7 | 7 - x^{2} | | 7 - x^{2} |) + 2 (x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))$ .

$f'' (x) = - \frac{\frac{2 (7 | 7 - x^{2} | | 7 - x^{2} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.2

Multiply $7 | 7 - x^{2} | | 7 - x^{2} |$ .

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Step 2.17.4.8.2.1

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = - \frac{\frac{2 (7 | (7 - x^{2}) (7 - x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.2.2

Raise $7 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 (7 | (7 - x^{2}) (7 - x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.2.3

Raise $7 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 (7 | (7 - x^{2}) (7 - x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.2.4

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

$f'' (x) = - \frac{\frac{2 (7 | {(7 - x^{2})}^{1 + 1} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.2.5

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{2 (7 | {(7 - x^{2})}^{2} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.3

Rewrite ${(7 - x^{2})}^{2}$ as $(7 - x^{2}) (7 - x^{2})$ .

$f'' (x) = - \frac{\frac{2 (7 | (7 - x^{2}) (7 - x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.4

Expand $(7 - x^{2}) (7 - x^{2})$ using the FOIL Method.

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Step 2.17.4.8.4.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (7 | 7 (7 - x^{2}) - x^{2} (7 - x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.4.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (7 | 7 \cdot 7 + 7 (- x^{2}) - x^{2} (7 - x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.4.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (7 | 7 \cdot 7 + 7 (- x^{2}) - x^{2} \cdot 7 - x^{2} (- x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5

Simplify and combine like terms.

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Step 2.17.4.8.5.1

Simplify each term.

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Step 2.17.4.8.5.1.1

Multiply $7$ by $7$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 + 7 (- x^{2}) - x^{2} \cdot 7 - x^{2} (- x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.2

Multiply $- 1$ by $7$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - x^{2} \cdot 7 - x^{2} (- x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.3

Multiply $7$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} - x^{2} (- x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{2} x^{2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.17.4.8.5.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 (x^{2} x^{2})) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.5.2

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

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{2 + 2}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} - 1 \cdot (- 1 x^{4}) | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} + 1 x^{4} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.1.7

Multiply $x^{4}$ by $1$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 7 x^{2} - 7 x^{2} + x^{4} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.5.2

Subtract $7 x^{2}$ from $- 7 x^{2}$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} | + 98 - 28 x^{2} + 2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.6

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | + x^{2} (- 3 | 49 - 14 x^{2} + x^{4} |) + x^{2} \cdot 98 + x^{2} (- 28 x^{2}) + x^{2} (2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.7

Simplify.

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Step 2.17.4.8.7.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + x^{2} \cdot 98 + x^{2} (- 28 x^{2}) + x^{2} (2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.7.2

Move $98$ to the left of $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 \cdot x^{2} + x^{2} (- 28 x^{2}) + x^{2} (2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.7.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 \cdot x^{2} - 28 x^{2} x^{2} + x^{2} (2 x^{4}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.7.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 \cdot x^{2} - 28 x^{2} x^{2} + 2 x^{2} x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.8

Simplify each term.

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Step 2.17.4.8.8.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.17.4.8.8.1.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 (x^{2} x^{2}) + 2 x^{2} x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.8.1.2

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

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{2 + 2} + 2 x^{2} x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.8.1.3

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{2} x^{4})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.8.2

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 2.17.4.8.8.2.1

Move $x^{4}$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 (x^{4} x^{2}))}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.8.2.2

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

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{4 + 2})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.4.8.8.2.3

Add $4$ and $2$ .

$f'' (x) = - \frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$

Step 2.17.5

Combine terms.

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Step 2.17.5.1

Rewrite $\frac{\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{| 7 - x^{2} |}}{{| 7 - x^{2} |}^{2}}$ as a product.

$f'' (x) = - (\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{| 7 - x^{2} |} \cdot \frac{1}{{| 7 - x^{2} |}^{2}})$

Step 2.17.5.2

Multiply $\frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{| 7 - x^{2} |}$ by $\frac{1}{{| 7 - x^{2} |}^{2}}$ .

$f'' (x) = - \frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{| 7 - x^{2} | {| 7 - x^{2} |}^{2}}$

Step 2.17.5.3

Multiply $| 7 - x^{2} |$ by ${| 7 - x^{2} |}^{2}$ by adding the exponents.

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Step 2.17.5.3.1

Multiply $| 7 - x^{2} |$ by ${| 7 - x^{2} |}^{2}$ .

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Step 2.17.5.3.1.1

Raise $| 7 - x^{2} |$ to the power of $1$ .

$f'' (x) = - \frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{| 7 - x^{2} | {| 7 - x^{2} |}^{2}}$

Step 2.17.5.3.1.2

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

$f'' (x) = - \frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{{| 7 - x^{2} |}^{1 + 2}}$

Step 2.17.5.3.2

Add $1$ and $2$ .

$f'' (x) = - \frac{2 (7 | 49 - 14 x^{2} + x^{4} | - 3 x^{2} | 49 - 14 x^{2} + x^{4} | + 98 x^{2} - 28 x^{4} + 2 x^{6})}{{| 7 - x^{2} |}^{3}}$

Step 3

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

$- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |} = 0$

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 7 - x^{2}$ .

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Step 4.1.1.1

To apply the Chain Rule, set $u$ as $7 - x^{2}$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [7 - x^{2}]$

Step 4.1.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [7 - x^{2}]$

Step 4.1.1.3

Replace all occurrences of $u$ with $7 - x^{2}$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} \frac{d}{d x} [7 - x^{2}]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $7 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- x^{2}]$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} (\frac{d}{d x} [7] + \frac{d}{d x} [- x^{2}])$

Step 4.1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} (0 + \frac{d}{d x} [- x^{2}])$

Step 4.1.2.3

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

$\frac{7 - x^{2}}{| 7 - x^{2} |} \frac{d}{d x} [- x^{2}]$

Step 4.1.2.4

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

$\frac{7 - x^{2}}{| 7 - x^{2} |} (- \frac{d}{d x} [x^{2}])$

Step 4.1.2.5

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

$\frac{7 - x^{2}}{| 7 - x^{2} |} (- (2 x))$

Step 4.1.2.6

Combine fractions.

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Step 4.1.2.6.1

Multiply $2$ by $- 1$ .

$\frac{7 - x^{2}}{| 7 - x^{2} |} (- 2 x)$

Step 4.1.2.6.2

Combine $- 2$ and $\frac{7 - x^{2}}{| 7 - x^{2} |}$ .

$\frac{- 2 (7 - x^{2})}{| 7 - x^{2} |} x$

Step 4.1.2.6.3

Combine $\frac{- 2 (7 - x^{2})}{| 7 - x^{2} |}$ and $x$ .

$\frac{- 2 (7 - x^{2}) x}{| 7 - x^{2} |}$

Step 4.1.2.6.4

Move the negative in front of the fraction.

$- \frac{(2 (7 - x^{2})) x}{| 7 - x^{2} |}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Apply the distributive property.

$- \frac{(2 \cdot 7 + 2 (- x^{2})) x}{| 7 - x^{2} |}$

Step 4.1.3.2

Apply the distributive property.

$- \frac{2 \cdot 7 x + 2 (- x^{2}) x}{| 7 - x^{2} |}$

Step 4.1.3.3

Simplify each term.

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Step 4.1.3.3.1

Multiply $2$ by $7$ .

$- \frac{14 x + 2 (- x^{2}) x}{| 7 - x^{2} |}$

Step 4.1.3.3.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.3.3.2.1

Move $x$ .

$- \frac{14 x + 2 (- (x \cdot x^{2}))}{| 7 - x^{2} |}$

Step 4.1.3.3.2.2

Multiply $x$ by $x^{2}$ .

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Step 4.1.3.3.2.2.1

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

$- \frac{14 x + 2 (- (x^{1} x^{2}))}{| 7 - x^{2} |}$

Step 4.1.3.3.2.2.2

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

$- \frac{14 x + 2 (- x^{1 + 2})}{| 7 - x^{2} |}$

Step 4.1.3.3.2.3

Add $1$ and $2$ .

$- \frac{14 x + 2 (- x^{3})}{| 7 - x^{2} |}$

Step 4.1.3.3.3

Multiply $- 1$ by $2$ .

$- \frac{14 x - 2 x^{3}}{| 7 - x^{2} |}$

Step 4.1.3.4

Factor $2 x$ out of $14 x - 2 x^{3}$ .

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Step 4.1.3.4.1

Factor $2 x$ out of $14 x$ .

$- \frac{2 x (7) - 2 x^{3}}{| 7 - x^{2} |}$

Step 4.1.3.4.2

Factor $2 x$ out of $- 2 x^{3}$ .

$- \frac{2 x (7) + 2 x (- x^{2})}{| 7 - x^{2} |}$

Step 4.1.3.4.3

Factor $2 x$ out of $2 x (7) + 2 x (- x^{2})$ .

$f' (x) = - \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ .

$- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |} = 0$

Step 5.2

Set the numerator equal to zero.

$2 x (7 - x^{2}) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$7 - x^{2} = 0$

Step 5.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.3

Set $7 - x^{2}$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $7 - x^{2}$ equal to $0$ .

$7 - x^{2} = 0$

Step 5.3.3.2

Solve $7 - x^{2} = 0$ for $x$ .

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Step 5.3.3.2.1

Subtract $7$ from both sides of the equation.

$- x^{2} = - 7$

Step 5.3.3.2.2

Divide each term in $- x^{2} = - 7$ by $- 1$ and simplify.

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Step 5.3.3.2.2.1

Divide each term in $- x^{2} = - 7$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 7}{- 1}$

Step 5.3.3.2.2.2

Simplify the left side.

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Step 5.3.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 7}{- 1}$

Step 5.3.3.2.2.2.2

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

$x^{2} = \frac{- 7}{- 1}$

Step 5.3.3.2.2.3

Simplify the right side.

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Step 5.3.3.2.2.3.1

Divide $- 7$ by $- 1$ .

$x^{2} = 7$

Step 5.3.3.2.3

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

$x = \pm \sqrt{7}$

Step 5.3.3.2.4

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

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Step 5.3.3.2.4.1

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

$x = \sqrt{7}$

Step 5.3.3.2.4.2

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

$x = - \sqrt{7}$

Step 5.3.3.2.4.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 5.3.4

The final solution is all the values that make $2 x (7 - x^{2}) = 0$ true.

$x = 0, \sqrt{7}, - \sqrt{7}$

Step 5.4

Exclude the solutions that do not make $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |} = 0$ true.

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ equal to $0$ to find where the expression is undefined.

$| 7 - x^{2} | = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$7 - x^{2} = \pm 0$

Step 6.2.2

Plus or minus $0$ is $0$ .

$7 - x^{2} = 0$

Step 6.2.3

Subtract $7$ from both sides of the equation.

$- x^{2} = - 7$

Step 6.2.4

Divide each term in $- x^{2} = - 7$ by $- 1$ and simplify.

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Step 6.2.4.1

Divide each term in $- x^{2} = - 7$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 7}{- 1}$

Step 6.2.4.2

Simplify the left side.

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Step 6.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 7}{- 1}$

Step 6.2.4.2.2

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

$x^{2} = \frac{- 7}{- 1}$

Step 6.2.4.3

Simplify the right side.

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Step 6.2.4.3.1

Divide $- 7$ by $- 1$ .

$x^{2} = 7$

Step 6.2.5

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

$x = \pm \sqrt{7}$

Step 6.2.6

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

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Step 6.2.6.1

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

$x = \sqrt{7}$

Step 6.2.6.2

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

$x = - \sqrt{7}$

Step 6.2.6.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 6.3

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 $0$ .

$x = - \sqrt{7}, x = \sqrt{7}$

Step 7

Critical points to evaluate.

$x = 0, - \sqrt{7}, \sqrt{7}$

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.

$- \frac{2 (7 | 49 - 14 {(0)}^{2} + {(0)}^{4} | - 3 {(0)}^{2} | 49 - 14 {(0)}^{2} + {(0)}^{4} | + 98 {(0)}^{2} - 28 {(0)}^{4} + 2 {(0)}^{6})}{{| 7 - {(0)}^{2} |}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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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 $0$ .

$- \frac{2 (7 | 49 - 14 \cdot 0 + 0^{4} | - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.1.2

Multiply $- 14$ by $0$ .

$- \frac{2 (7 | 49 + 0 + 0^{4} | - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.1.3

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

$- \frac{2 (7 | 49 + 0 + 0 | - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.2

Add $49$ and $0$ .

$- \frac{2 (7 | 49 + 0 | - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.3

Add $49$ and $0$ .

$- \frac{2 (7 | 49 | - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.4

The absolute value is the distance between a number and zero. The distance between $0$ and $49$ is $49$ .

$- \frac{2 (7 \cdot 49 - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.5

Multiply $7$ by $49$ .

$- \frac{2 (343 - 3 \cdot 0^{2} | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.6

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

$- \frac{2 (343 - 3 \cdot 0 | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.7

Multiply $- 3$ by $0$ .

$- \frac{2 (343 + 0 | 49 - 14 \cdot 0^{2} + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.8

Simplify each term.

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Step 9.1.8.1

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

$- \frac{2 (343 + 0 | 49 - 14 \cdot 0 + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.8.2

Multiply $- 14$ by $0$ .

$- \frac{2 (343 + 0 | 49 + 0 + 0^{4} | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.8.3

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

$- \frac{2 (343 + 0 | 49 + 0 + 0 | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.9

Add $49$ and $0$ .

$- \frac{2 (343 + 0 | 49 + 0 | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.10

Add $49$ and $0$ .

$- \frac{2 (343 + 0 | 49 | + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.11

The absolute value is the distance between a number and zero. The distance between $0$ and $49$ is $49$ .

$- \frac{2 (343 + 0 \cdot 49 + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.12

Multiply $0$ by $49$ .

$- \frac{2 (343 + 0 + 98 \cdot 0^{2} - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.13

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

$- \frac{2 (343 + 0 + 98 \cdot 0 - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.14

Multiply $98$ by $0$ .

$- \frac{2 (343 + 0 + 0 - 28 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.15

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

$- \frac{2 (343 + 0 + 0 - 28 \cdot 0 + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.16

Multiply $- 28$ by $0$ .

$- \frac{2 (343 + 0 + 0 + 0 + 2 \cdot 0^{6})}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.17

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

$- \frac{2 (343 + 0 + 0 + 0 + 2 \cdot 0)}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.18

Multiply $2$ by $0$ .

$- \frac{2 (343 + 0 + 0 + 0 + 0)}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.19

Add $343 + 0 + 0 + 0$ and $0$ .

$- \frac{2 (343 + 0 + 0 + 0)}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.20

Add $343 + 0 + 0$ and $0$ .

$- \frac{2 (343 + 0 + 0)}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.21

Add $343 + 0$ and $0$ .

$- \frac{2 (343 + 0)}{{| 7 - 0^{2} |}^{3}}$

Step 9.1.22

Add $343$ and $0$ .

$- \frac{2 \cdot 343}{{| 7 - 0^{2} |}^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$- \frac{2 \cdot 343}{{| 7 - 0 |}^{3}}$

Step 9.2.2

Multiply $- 1$ by $0$ .

$- \frac{2 \cdot 343}{{| 7 + 0 |}^{3}}$

Step 9.2.3

Add $7$ and $0$ .

$- \frac{2 \cdot 343}{{| 7 |}^{3}}$

Step 9.2.4

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$- \frac{2 \cdot 343}{7^{3}}$

Step 9.2.5

Raise $7$ to the power of $3$ .

$- \frac{2 \cdot 343}{343}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Multiply $2$ by $343$ .

$- \frac{686}{343}$

Step 9.3.2

Divide $686$ by $343$ .

$- 1 \cdot 2$

Step 9.3.3

Multiply $- 1$ by $2$ .

$- 2$

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.

$f (0) = | 7 - {(0)}^{2} |$

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

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

$f (0) = | 7 - 0 |$

Step 11.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = | 7 + 0 |$

Step 11.2.2

Add $7$ and $0$ .

$f (0) = | 7 |$

Step 11.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$f (0) = 7$

Step 11.2.4

The final answer is $7$ .

$y = 7$

Step 12

Evaluate the second derivative at $x = - \sqrt{7}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (7 | 49 - 14 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{4} | - 3 {(- \sqrt{7})}^{2} | 49 - 14 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{4} | + 98 {(- \sqrt{7})}^{2} - 28 {(- \sqrt{7})}^{4} + 2 {(- \sqrt{7})}^{6})}{{| 7 - {(- \sqrt{7})}^{2} |}^{3}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the product rule to $- \sqrt{7}$ .

Step 13.1.2

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

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Step 13.1.2.1

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

Step 13.1.2.2

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

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Step 13.1.2.2.1

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

Step 13.1.2.2.2

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

Step 13.1.2.3

Add $2$ and $1$ .

Step 13.1.3

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

Step 13.1.4

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 13.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

Step 13.1.4.2

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

Step 13.1.4.3

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

Step 13.1.4.4

Cancel the common factor of $2$ .

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Step 13.1.4.4.1

Cancel the common factor.

Step 13.1.4.4.2

Rewrite the expression.

Step 13.1.4.5

Evaluate the exponent.

Step 13.1.5

Multiply $- 1$ by $7$ .

Step 13.2

Subtract $7$ from $7$ .

Step 13.3

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

Step 13.4

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

Step 13.5

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

Undefined

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - \sqrt{7}) \cup (- \sqrt{7}, 0) \cup (0, \sqrt{7}) \cup (\sqrt{7}, \infty)$

Step 14.2

Substitute any number, such as $- 5$ , from the interval $(- \infty, - \sqrt{7})$ in the first derivative $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = - \frac{2 (- 5) (7 - {(- 5)}^{2})}{| 7 - {(- 5)}^{2} |}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Multiply $2$ by $- 5$ .

$f' (- 5) = - \frac{- 10 (7 - {(- 5)}^{2})}{| 7 - {(- 5)}^{2} |}$

Step 14.2.2.2

Simplify the denominator.

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Step 14.2.2.2.1

Raise $- 5$ to the power of $2$ .

$f' (- 5) = - \frac{- 10 (7 - {(- 5)}^{2})}{| 7 - 1 \cdot 25 |}$

Step 14.2.2.2.2

Multiply $- 1$ by $25$ .

$f' (- 5) = - \frac{- 10 (7 - {(- 5)}^{2})}{| 7 - 25 |}$

Step 14.2.2.2.3

Subtract $25$ from $7$ .

$f' (- 5) = - \frac{- 10 (7 - {(- 5)}^{2})}{| - 18 |}$

Step 14.2.2.2.4

The absolute value is the distance between a number and zero. The distance between $- 18$ and $0$ is $18$ .

$f' (- 5) = - \frac{- 10 (7 - {(- 5)}^{2})}{18}$

Step 14.2.2.3

Simplify the numerator.

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Step 14.2.2.3.1

Raise $- 5$ to the power of $2$ .

$f' (- 5) = - \frac{- 10 (7 - 1 \cdot 25)}{18}$

Step 14.2.2.3.2

Multiply $- 1$ by $25$ .

$f' (- 5) = - \frac{- 10 (7 - 25)}{18}$

Step 14.2.2.3.3

Subtract $25$ from $7$ .

$f' (- 5) = - \frac{- 10 \cdot - 18}{18}$

Step 14.2.2.4

Simplify the expression.

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Step 14.2.2.4.1

Multiply $- 10$ by $- 18$ .

$f' (- 5) = - \frac{180}{18}$

Step 14.2.2.4.2

Divide $180$ by $18$ .

$f' (- 5) = - 1 \cdot 10$

Step 14.2.2.4.3

Multiply $- 1$ by $10$ .

$f' (- 5) = - 10$

Step 14.2.2.5

The final answer is $- 10$ .

$- 10$

Step 14.3

Substitute any number, such as $- 1$ , from the interval $(- \sqrt{7}, 0)$ in the first derivative $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{2 (- 1) (7 - {(- 1)}^{2})}{| 7 - {(- 1)}^{2} |}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

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

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Step 14.3.2.1.1

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

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Step 14.3.2.1.1.1

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

$f' (- 1) = - \frac{2 (- 1) (7 + (- 1) {(- 1)}^{2})}{| 7 - {(- 1)}^{2} |}$

Step 14.3.2.1.1.2

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

$f' (- 1) = - \frac{2 (- 1) (7 + {(- 1)}^{1 + 2})}{| 7 - {(- 1)}^{2} |}$

Step 14.3.2.1.2

Add $1$ and $2$ .

$f' (- 1) = - \frac{2 (- 1) (7 + {(- 1)}^{3})}{| 7 - {(- 1)}^{2} |}$

Step 14.3.2.2

Multiply $2$ by $- 1$ .

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{| 7 - {(- 1)}^{2} |}$

Step 14.3.2.3

Simplify the denominator.

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Step 14.3.2.3.1

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

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Step 14.3.2.3.1.1

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

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Step 14.3.2.3.1.1.1

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

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{| 7 + (- 1) {(- 1)}^{2} |}$

Step 14.3.2.3.1.1.2

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

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{| 7 + {(- 1)}^{1 + 2} |}$

Step 14.3.2.3.1.2

Add $1$ and $2$ .

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{| 7 + {(- 1)}^{3} |}$

Step 14.3.2.3.2

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

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{| 7 - 1 |}$

Step 14.3.2.3.3

Subtract $1$ from $7$ .

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{| 6 |}$

Step 14.3.2.3.4

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$f' (- 1) = - \frac{- 2 (7 + {(- 1)}^{3})}{6}$

Step 14.3.2.4

Simplify the numerator.

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Step 14.3.2.4.1

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

$f' (- 1) = - \frac{- 2 (7 - 1)}{6}$

Step 14.3.2.4.2

Subtract $1$ from $7$ .

$f' (- 1) = - \frac{- 2 \cdot 6}{6}$

Step 14.3.2.5

Simplify the expression.

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Step 14.3.2.5.1

Multiply $- 2$ by $6$ .

$f' (- 1) = - \frac{- 12}{6}$

Step 14.3.2.5.2

Divide $- 12$ by $6$ .

$f' (- 1) = 2$

Step 14.3.2.5.3

Multiply $- 1$ by $- 2$ .

$f' (- 1) = 2$

Step 14.3.2.6

The final answer is $2$ .

$2$

Step 14.4

Substitute any number, such as $1$ , from the interval $(0, \sqrt{7})$ in the first derivative $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = - \frac{2 (1) (7 - {(1)}^{2})}{| 7 - {(1)}^{2} |}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Multiply $2$ by $1$ .

$f' (1) = - \frac{2 (7 - 1^{2})}{| 7 - 1^{2} |}$

Step 14.4.2.2

Simplify the denominator.

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Step 14.4.2.2.1

One to any power is one.

$f' (1) = - \frac{2 (7 - 1^{2})}{| 7 - 1 \cdot 1 |}$

Step 14.4.2.2.2

Multiply $- 1$ by $1$ .

$f' (1) = - \frac{2 (7 - 1^{2})}{| 7 - 1 |}$

Step 14.4.2.2.3

Subtract $1$ from $7$ .

$f' (1) = - \frac{2 (7 - 1^{2})}{| 6 |}$

Step 14.4.2.2.4

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$f' (1) = - \frac{2 (7 - 1^{2})}{6}$

Step 14.4.2.3

Simplify the numerator.

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Step 14.4.2.3.1

One to any power is one.

$f' (1) = - \frac{2 (7 - 1 \cdot 1)}{6}$

Step 14.4.2.3.2

Multiply $- 1$ by $1$ .

$f' (1) = - \frac{2 (7 - 1)}{6}$

Step 14.4.2.3.3

Subtract $1$ from $7$ .

$f' (1) = - \frac{2 \cdot 6}{6}$

Step 14.4.2.4

Simplify the expression.

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Step 14.4.2.4.1

Multiply $2$ by $6$ .

$f' (1) = - \frac{12}{6}$

Step 14.4.2.4.2

Divide $12$ by $6$ .

$f' (1) = - 1 \cdot 2$

Step 14.4.2.4.3

Multiply $- 1$ by $2$ .

$f' (1) = - 2$

Step 14.4.2.5

The final answer is $- 2$ .

$- 2$

Step 14.5

Substitute any number, such as $5$ , from the interval $(\sqrt{7}, \infty)$ in the first derivative $- \frac{2 x (7 - x^{2})}{| 7 - x^{2} |}$ to check if the result is negative or positive.

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Step 14.5.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = - \frac{2 (5) (7 - {(5)}^{2})}{| 7 - {(5)}^{2} |}$

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Multiply $2$ by $5$ .

$f' (5) = - \frac{10 (7 - 5^{2})}{| 7 - 5^{2} |}$

Step 14.5.2.2

Simplify the denominator.

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Step 14.5.2.2.1

Raise $5$ to the power of $2$ .

$f' (5) = - \frac{10 (7 - 5^{2})}{| 7 - 1 \cdot 25 |}$

Step 14.5.2.2.2

Multiply $- 1$ by $25$ .

$f' (5) = - \frac{10 (7 - 5^{2})}{| 7 - 25 |}$

Step 14.5.2.2.3

Subtract $25$ from $7$ .

$f' (5) = - \frac{10 (7 - 5^{2})}{| - 18 |}$

Step 14.5.2.2.4

The absolute value is the distance between a number and zero. The distance between $- 18$ and $0$ is $18$ .

$f' (5) = - \frac{10 (7 - 5^{2})}{18}$

Step 14.5.2.3

Simplify the numerator.

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Step 14.5.2.3.1

Raise $5$ to the power of $2$ .

$f' (5) = - \frac{10 (7 - 1 \cdot 25)}{18}$

Step 14.5.2.3.2

Multiply $- 1$ by $25$ .

$f' (5) = - \frac{10 (7 - 25)}{18}$

Step 14.5.2.3.3

Subtract $25$ from $7$ .

$f' (5) = - \frac{10 \cdot - 18}{18}$

Step 14.5.2.4

Simplify the expression.

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Step 14.5.2.4.1

Multiply $10$ by $- 18$ .

$f' (5) = - \frac{- 180}{18}$

Step 14.5.2.4.2

Divide $- 180$ by $18$ .

$f' (5) = 10$

Step 14.5.2.4.3

Multiply $- 1$ by $- 10$ .

$f' (5) = 10$

Step 14.5.2.5

The final answer is $10$ .

$10$

Step 14.6

Since the first derivative changed signs from negative to positive around $x = - \sqrt{7}$ , then $x = - \sqrt{7}$ is a local minimum.

$x = - \sqrt{7}$ is a local minimum

Step 14.7

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 14.8

Since the first derivative changed signs from negative to positive around $x = \sqrt{7}$ , then $x = \sqrt{7}$ is a local minimum.

$x = \sqrt{7}$ is a local minimum

Step 14.9

These are the local extrema for $f (x) = | 7 - x^{2} |$ .

$x = - \sqrt{7}$ is a local minimum

$x = 0$ is a local maximum

$x = \sqrt{7}$ is a local minimum

$x = - \sqrt{7}$ is a local minimum

$x = 0$ is a local maximum

$x = \sqrt{7}$ is a local minimum

Step 15