Calculus Examples

$f (x) = | x^{2} - 100 |$

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 $g (x) = x^{2} - 100$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 100$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - 100]$

Step 1.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [x^{2} - 100]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} - 100$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} \frac{d}{d x} [x^{2} - 100]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 100$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 100]$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 100])$

Step 1.2.2

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

$\frac{x^{2} - 100}{| x^{2} - 100 |} (2 x + \frac{d}{d x} [- 100])$

Step 1.2.3

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100$ with respect to $x$ is $0$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} (2 x + 0)$

Step 1.2.4

Combine fractions.

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Step 1.2.4.1

Add $2 x$ and $0$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} (2 x)$

Step 1.2.4.2

Combine $2$ and $\frac{x^{2} - 100}{| x^{2} - 100 |}$ .

$\frac{2 (x^{2} - 100)}{| x^{2} - 100 |} x$

Step 1.2.4.3

Combine $\frac{2 (x^{2} - 100)}{| x^{2} - 100 |}$ and $x$ .

$\frac{2 (x^{2} - 100) x}{| x^{2} - 100 |}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot - 100) x}{| x^{2} - 100 |}$

Step 1.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 1.3.3

Simplify each term.

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Step 1.3.3.1

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

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Step 1.3.3.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 1.3.3.1.2

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

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Step 1.3.3.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 1.3.3.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 1.3.3.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 1.3.3.2

Multiply $2$ by $- 100$ .

$\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x^{3} - 200 x$ and $g (x) = | x^{2} - 100 |$ .

$f'' (x) = \frac{| x^{2} - 100 | \frac{d}{d x} (2 x^{3} - 200 x) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $2 x^{3} - 200 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 200 x]$ .

$f'' (x) = \frac{| x^{2} - 100 | (\frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (- 200 x)) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = \frac{| x^{2} - 100 | (2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 200 x)) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = \frac{| x^{2} - 100 | (2 (3 x^{2}) + \frac{d}{d x} (- 200 x)) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2.4

Multiply $3$ by $2$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} + \frac{d}{d x} (- 200 x)) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2.5

Since $- 200$ is constant with respect to $x$ , the derivative of $- 200 x$ with respect to $x$ is $- 200 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200 \frac{d}{d x} x) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2.6

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

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200 \cdot 1) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.2.7

Multiply $- 200$ by $1$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) \frac{d}{d x} | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 100$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 100$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{d}{d u} (| u |) \frac{d}{d x} (x^{2} - 100))}{{| x^{2} - 100 |}^{2}}$

Step 2.3.2

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

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{u}{| u |} \cdot \frac{d}{d x} (x^{2} - 100))}{{| x^{2} - 100 |}^{2}}$

Step 2.3.3

Replace all occurrences of $u$ with $x^{2} - 100$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{x^{2} - 100}{| x^{2} - 100 |} \cdot \frac{d}{d x} (x^{2} - 100))}{{| x^{2} - 100 |}^{2}}$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $x^{2} - 100$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 100]$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{x^{2} - 100}{| x^{2} - 100 |} \cdot (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 100)))}{{| x^{2} - 100 |}^{2}}$

Step 2.4.2

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

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{x^{2} - 100}{| x^{2} - 100 |} \cdot (2 x + \frac{d}{d x} (- 100)))}{{| x^{2} - 100 |}^{2}}$

Step 2.4.3

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100$ with respect to $x$ is $0$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{x^{2} - 100}{| x^{2} - 100 |} \cdot (2 x + 0))}{{| x^{2} - 100 |}^{2}}$

Step 2.4.4

Combine fractions.

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Step 2.4.4.1

Add $2 x$ and $0$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{x^{2} - 100}{| x^{2} - 100 |} \cdot (2 x))}{{| x^{2} - 100 |}^{2}}$

Step 2.4.4.2

Combine $2$ and $\frac{x^{2} - 100}{| x^{2} - 100 |}$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) (\frac{2 (x^{2} - 100)}{| x^{2} - 100 |} \cdot x)}{{| x^{2} - 100 |}^{2}}$

Step 2.4.4.3

Combine $\frac{2 (x^{2} - 100)}{| x^{2} - 100 |}$ and $x$ .

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) - (2 x^{3} - 200 x) \frac{2 (x^{2} - 100) x}{| x^{2} - 100 |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) + (- (2 x^{3}) - (- 200 x)) (\frac{2 (x^{2} - 100) x}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.2

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) + (- (2 x^{3}) - (- 200 x)) (\frac{(2 x^{2} + 2 \cdot - 100) x}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.3

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2} - 200) + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{2} x + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Simplify each term.

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Step 2.5.4.1.1

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 100 | (6 x^{2}) + | x^{2} - 100 | \cdot - 200 + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{2} x + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} + | x^{2} - 100 | \cdot - 200 + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{2} x + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.3

Move $- 200$ to the left of $| x^{2} - 100 |$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 \cdot | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{2} x + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4

Simplify the numerator.

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Step 2.5.4.1.4.1

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

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Step 2.5.4.1.4.1.1

Move $x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 (x \cdot x^{2}) + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.1.2

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

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Step 2.5.4.1.4.1.2.1

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 (x \cdot x^{2}) + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.1.2.2

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{1 + 2} + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{3} + 2 \cdot (- 100 x)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.2

Multiply $2$ by $- 100$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x^{3} - 200 x}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.3

Rewrite $2 x^{3} - 200 x$ in a factored form.

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Step 2.5.4.1.4.3.1

Factor $2 x$ out of $2 x^{3} - 200 x$ .

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Step 2.5.4.1.4.3.1.1

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x^{2}) - 200 x}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.3.1.2

Factor $2 x$ out of $- 200 x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x^{2}) + 2 x (- 100)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.3.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 100)$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x^{2} - 100)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.3.2

Rewrite $100$ as $10^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x^{2} - 10^{2})}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.4.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 10$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5

Simplify the denominator.

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Step 2.5.4.1.5.1

Rewrite $100$ as $10^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x^{2} - 10^{2} |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 10$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| (x + 10) (x - 10) |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.3

Expand $(x + 10) (x - 10)$ using the FOIL Method.

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Step 2.5.4.1.5.3.1

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x (x - 10) + 10 (x - 10) |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.3.2

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x \cdot x + x \cdot - 10 + 10 (x - 10) |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.3.3

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.4

Combine the opposite terms in $x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10$ .

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Step 2.5.4.1.5.4.1

Reorder the factors in the terms $x \cdot - 10$ and $10 x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x \cdot x - 10 x + 10 x + 10 \cdot - 10 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.4.2

Add $- 10 x$ and $10 x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x \cdot x + 0 + 10 \cdot - 10 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.4.3

Add $x \cdot x$ and $0$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x \cdot x + 10 \cdot - 10 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.5

Simplify each term.

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Step 2.5.4.1.5.5.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x^{2} + 10 \cdot - 10 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.5.2

Multiply $10$ by $- 10$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x^{2} - 100 |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.6

Rewrite $100$ as $10^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| x^{2} - 10^{2} |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.5.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 10$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- (2 x^{3}) - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| (x + 10) (x - 10) |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.6

Simplify each term.

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Step 2.5.4.1.6.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- 2 x^{3} - (- 200 x)) (\frac{2 x (x + 10) (x - 10)}{| (x + 10) (x - 10) |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.6.2

Multiply $- 200$ by $- 1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + (- 2 x^{3} + 200 x) (\frac{2 x (x + 10) (x - 10)}{| (x + 10) (x - 10) |})}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.7

Multiply $- 2 x^{3} + 200 x$ by $\frac{2 x (x + 10) (x - 10)}{| (x + 10) (x - 10) |}$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{(- 2 x^{3} + 200 x) (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8

Simplify the numerator.

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Step 2.5.4.1.8.1

Factor $2 x$ out of $- 2 x^{3} + 200 x$ .

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Step 2.5.4.1.8.1.1

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{(2 x (- x^{2}) + 200 x) \cdot (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.1.2

Factor $2 x$ out of $200 x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{(2 x (- x^{2}) + 2 x (100)) \cdot (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.1.3

Factor $2 x$ out of $2 x (- x^{2}) + 2 x (100)$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{2 x (- x^{2} + 100) \cdot (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.2

Rewrite $100$ as $10^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{2 x (- x^{2} + 10^{2}) \cdot (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.3

Reorder $- x^{2}$ and $10^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{2 x (10^{2} - x^{2}) \cdot (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{2 x (10 + x) (10 - x) \cdot (2 x (x + 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5

Combine exponents.

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Step 2.5.4.1.8.5.1

Multiply $2$ by $2$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x (10 + x) (10 - x) x (x + 10) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.2

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 (x \cdot x) (10 + x) (10 - x) (x + 10) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.3

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 (x \cdot x) (10 + x) (10 - x) (x + 10) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.4

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{1 + 1} (10 + x) (10 - x) (x + 10) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.5

Add $1$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} (10 + x) (10 - x) (x + 10) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.6

Reorder terms.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} (x + 10) (10 - x) (x + 10) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.7

Raise $x + 10$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} ((x + 10) (x + 10)) (10 - x) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.8

Raise $x + 10$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} ((x + 10) (x + 10)) (10 - x) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.9

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{1 + 1} (10 - x) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.10

Add $1$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} (10 - x) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.11

Rewrite $10$ as $- 1 (- 10)$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} (- 1 \cdot - 10 - x) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.12

Factor $- 1$ out of $- x$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} (- 1 \cdot - 10 - (x)) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.13

Factor $- 1$ out of $- 1 (- 10) - (x)$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} (- 1 (- 10 + x)) (x - 10)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.14

Reorder terms.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} \cdot (- 1 (x - 10) (x - 10))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.15

Raise $x - 10$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} \cdot (- 1 ((x - 10) (x - 10)))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.16

Raise $x - 10$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} \cdot (- 1 ((x - 10) (x - 10)))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.17

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

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} \cdot (- 1 {(x - 10)}^{1 + 1})}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.18

Add $1$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{4 x^{2} {(x + 10)}^{2} \cdot (- 1 {(x - 10)}^{2})}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.8.5.19

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | + \frac{- 4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.1.9

Move the negative in front of the fraction.

$f'' (x) = \frac{6 | x^{2} - 100 | x^{2} - 200 | x^{2} - 100 | - \frac{4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.2

To write $6 | x^{2} - 100 | x^{2}$ as a fraction with a common denominator, multiply by $\frac{| (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}$ .

$f'' (x) = \frac{\frac{6 | x^{2} - 100 | x^{2} | (x + 10) (x - 10) |}{| (x + 10) (x - 10) |} - \frac{4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{6 | x^{2} - 100 | x^{2} | (x + 10) (x - 10) | - 4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4

Simplify each term.

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Step 2.5.4.4.1

Simplify the numerator.

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Step 2.5.4.4.1.1

Factor $2 x^{2}$ out of $6 | x^{2} - 100 | x^{2} | (x + 10) (x - 10) | - 4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}$ .

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Step 2.5.4.4.1.1.1

Factor $2 x^{2}$ out of $6 | x^{2} - 100 | x^{2} | (x + 10) (x - 10) |$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2} - 100 | | (x + 10) (x - 10) |) - 4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.1.2

Factor $2 x^{2}$ out of $- 4 x^{2} {(x + 10)}^{2} {(x - 10)}^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2} - 100 | | (x + 10) (x - 10) |) + 2 x^{2} (- 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.1.3

Factor $2 x^{2}$ out of $2 x^{2} (3 | x^{2} - 100 | | (x + 10) (x - 10) |) + 2 x^{2} (- 2 {(x + 10)}^{2} {(x - 10)}^{2})$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2} - 100 | | (x + 10) (x - 10) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.2

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

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x + 10) (x - 10) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3

Simplify each term.

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Step 2.5.4.4.1.3.1

Expand $(x + 10) (x - 10)$ using the FOIL Method.

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Step 2.5.4.4.1.3.1.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x (x - 10) + 10 (x - 10)) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.1.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x \cdot x + x \cdot - 10 + 10 (x - 10)) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.1.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.2

Combine the opposite terms in $x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10$ .

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Step 2.5.4.4.1.3.2.1

Reorder the factors in the terms $x \cdot - 10$ and $10 x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x \cdot x - 10 x + 10 x + 10 \cdot - 10) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.2.2

Add $- 10 x$ and $10 x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x \cdot x + 0 + 10 \cdot - 10) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.2.3

Add $x \cdot x$ and $0$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x \cdot x + 10 \cdot - 10) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.3

Simplify each term.

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Step 2.5.4.4.1.3.3.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x^{2} + 10 \cdot - 10) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.3.2

Multiply $10$ by $- 10$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | (x^{2} - 100) (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.4

Expand $(x^{2} - 100) (x^{2} - 100)$ using the FOIL Method.

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Step 2.5.4.4.1.3.4.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2} (x^{2} - 100) - 100 (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.4.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2} x^{2} + x^{2} \cdot - 100 - 100 (x^{2} - 100) | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.4.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2} x^{2} + x^{2} \cdot - 100 - 100 x^{2} - 100 \cdot - 100 | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.5

Simplify and combine like terms.

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Step 2.5.4.4.1.3.5.1

Simplify each term.

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Step 2.5.4.4.1.3.5.1.1

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

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Step 2.5.4.4.1.3.5.1.1.1

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

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{2 + 2} + x^{2} \cdot - 100 - 100 x^{2} - 100 \cdot - 100 | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.5.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} + x^{2} \cdot - 100 - 100 x^{2} - 100 \cdot - 100 | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.5.1.2

Move $- 100$ to the left of $x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 100 \cdot x^{2} - 100 x^{2} - 100 \cdot - 100 | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.5.1.3

Multiply $- 100$ by $- 100$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 100 x^{2} - 100 x^{2} + 10000 | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.5.2

Subtract $100 x^{2}$ from $- 100 x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 {(x + 10)}^{2} {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.6

Rewrite ${(x + 10)}^{2}$ as $(x + 10) (x + 10)$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 ((x + 10) (x + 10)) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.7

Expand $(x + 10) (x + 10)$ using the FOIL Method.

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Step 2.5.4.4.1.3.7.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x (x + 10) + 10 (x + 10)) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.7.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x \cdot x + x \cdot 10 + 10 (x + 10)) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.7.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x \cdot x + x \cdot 10 + 10 x + 10 \cdot 10) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.8

Simplify and combine like terms.

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Step 2.5.4.4.1.3.8.1

Simplify each term.

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Step 2.5.4.4.1.3.8.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x^{2} + x \cdot 10 + 10 x + 10 \cdot 10) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.8.1.2

Move $10$ to the left of $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x^{2} + 10 \cdot x + 10 x + 10 \cdot 10) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.8.1.3

Multiply $10$ by $10$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x^{2} + 10 x + 10 x + 100) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.8.2

Add $10 x$ and $10 x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x^{2} + 20 x + 100) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.9

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 2 (20 x) - 2 \cdot 100) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.10

Simplify.

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Step 2.5.4.4.1.3.10.1

Multiply $20$ by $- 2$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 2 \cdot 100) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.10.2

Multiply $- 2$ by $100$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) {(x - 10)}^{2})}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.11

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) ((x - 10) (x - 10)))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.12

Expand $(x - 10) (x - 10)$ using the FOIL Method.

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Step 2.5.4.4.1.3.12.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x (x - 10) - 10 (x - 10)))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.12.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x \cdot x + x \cdot - 10 - 10 (x - 10)))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.12.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.13

Simplify and combine like terms.

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Step 2.5.4.4.1.3.13.1

Simplify each term.

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Step 2.5.4.4.1.3.13.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.13.1.2

Move $- 10$ to the left of $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.13.1.3

Multiply $- 10$ by $- 10$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x^{2} - 10 x - 10 x + 100))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.13.2

Subtract $10 x$ from $- 10 x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | + (- 2 x^{2} - 40 x - 200) (x^{2} - 20 x + 100))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.14

Expand $(- 2 x^{2} - 40 x - 200) (x^{2} - 20 x + 100)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{2} x^{2} - 2 x^{2} (- 20 x) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15

Simplify each term.

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Step 2.5.4.4.1.3.15.1

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

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Step 2.5.4.4.1.3.15.1.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 (x^{2} x^{2}) - 2 x^{2} (- 20 x) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.1.2

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

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{2 + 2} - 2 x^{2} (- 20 x) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.1.3

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 2 x^{2} (- 20 x) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 2 \cdot (- 20 x^{2} x) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.3

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

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Step 2.5.4.4.1.3.15.3.1

Move $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 2 \cdot (- 20 (x \cdot x^{2})) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.3.2

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

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Step 2.5.4.4.1.3.15.3.2.1

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

Step 2.5.4.4.1.3.15.3.2.2

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

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 2 \cdot (- 20 x^{1 + 2}) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 2 \cdot (- 20 x^{3}) - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.4

Multiply $- 2$ by $- 20$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 2 x^{2} \cdot 100 - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.5

Multiply $100$ by $- 2$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x \cdot x^{2} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.6

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

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Step 2.5.4.4.1.3.15.6.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 (x^{2} x) - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.6.2

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

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Step 2.5.4.4.1.3.15.6.2.1

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

Step 2.5.4.4.1.3.15.6.2.2

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

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{2 + 1} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.6.3

Add $2$ and $1$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} - 40 x (- 20 x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} - 40 \cdot (- 20 x \cdot x) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.8

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.4.4.1.3.15.8.1

Move $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} - 40 \cdot (- 20 (x \cdot x)) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.8.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} - 40 \cdot (- 20 x^{2}) - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.9

Multiply $- 40$ by $- 20$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} + 800 x^{2} - 40 x \cdot 100 - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.10

Multiply $100$ by $- 40$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} + 800 x^{2} - 4000 x - 200 x^{2} - 200 (- 20 x) - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.11

Multiply $- 20$ by $- 200$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} + 800 x^{2} - 4000 x - 200 x^{2} + 4000 x - 200 \cdot 100)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.15.12

Multiply $- 200$ by $100$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} + 800 x^{2} - 4000 x - 200 x^{2} + 4000 x - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.16

Combine the opposite terms in $- 2 x^{4} + 40 x^{3} - 200 x^{2} - 40 x^{3} + 800 x^{2} - 4000 x - 200 x^{2} + 4000 x - 20000$ .

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Step 2.5.4.4.1.3.16.1

Subtract $40 x^{3}$ from $40 x^{3}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 200 x^{2} + 0 + 800 x^{2} - 4000 x - 200 x^{2} + 4000 x - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.16.2

Add $- 2 x^{4} - 200 x^{2}$ and $0$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 200 x^{2} + 800 x^{2} - 4000 x - 200 x^{2} + 4000 x - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.16.3

Add $- 4000 x$ and $4000 x$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 200 x^{2} + 800 x^{2} - 200 x^{2} + 0 - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.16.4

Add $- 2 x^{4} - 200 x^{2} + 800 x^{2} - 200 x^{2}$ and $0$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} - 200 x^{2} + 800 x^{2} - 200 x^{2} - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.17

Add $- 200 x^{2}$ and $800 x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 600 x^{2} - 200 x^{2} - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.3.18

Subtract $200 x^{2}$ from $600 x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (3 | x^{4} - 200 x^{2} + 10000 | - 2 x^{4} + 400 x^{2} - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.4

Reorder terms.

$f'' (x) = \frac{\frac{2 x^{2} (- 2 x^{4} + 400 x^{2} + 3 | x^{4} - 200 x^{2} + 10000 | - 20000)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5

Rewrite $- 2 x^{4} + 400 x^{2} + 3 | x^{4} - 200 x^{2} + 10000 | - 20000$ in a factored form.

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Step 2.5.4.4.1.5.1

Regroup terms.

$f'' (x) = \frac{\frac{2 x^{2} (- 20000 - 2 x^{4} + 400 x^{2} + 3 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.2

Factor $- 1$ out of $- 2 x^{4} + 400 x^{2} + 3 | x^{4} - 200 x^{2} + 10000 |$ .

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Step 2.5.4.4.1.5.2.1

Factor $- 1$ out of $- 2 x^{4}$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 20000 - (2 x^{4}) + 400 x^{2} + 3 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.2.2

Factor $- 1$ out of $400 x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 20000 - (2 x^{4}) - (- 400 x^{2}) + 3 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.2.3

Factor $- 1$ out of $3 | x^{4} - 200 x^{2} + 10000 |$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 20000 - (2 x^{4}) - (- 400 x^{2}) - (- 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.2.4

Factor $- 1$ out of $- (2 x^{4}) - (- 400 x^{2})$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 20000 - (2 x^{4} - 400 x^{2}) - (- 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.2.5

Factor $- 1$ out of $- (2 x^{4} - 400 x^{2}) - (- 3 | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 20000 - (2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.3

Factor $- 1$ out of $- 20000 - (2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |)$ .

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Step 2.5.4.4.1.5.3.1

Rewrite $- 20000$ as $- 1 (20000)$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 1 \cdot 20000 - (2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.3.2

Factor $- 1$ out of $- 1 (20000) - (2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 x^{2} (- 1 (20000 + 2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.3.3

Rewrite $- 1 (20000 + 2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |)$ as $- (20000 + 2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 x^{2} (- (20000 + 2 x^{4} - 400 x^{2} - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.5.4

Reorder terms.

$f'' (x) = \frac{\frac{2 x^{2} (- (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

$f'' (x) = \frac{\frac{2 x^{2} \cdot (- 1 (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.6

Combine exponents.

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Step 2.5.4.4.1.6.1

Factor out negative.

$f'' (x) = \frac{\frac{- (2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.1.6.2

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{\frac{- 2 (x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.4.2

Move the negative in front of the fraction.

$f'' (x) = \frac{- \frac{2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 |}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.5

To write $- 200 | x^{2} - 100 |$ as a fraction with a common denominator, multiply by $\frac{| (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}$ .

$f'' (x) = \frac{- \frac{2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} - 200 | x^{2} - 100 | \cdot \frac{| (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.6

Combine $- 200 | x^{2} - 100 |$ and $\frac{| (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}$ .

$f'' (x) = \frac{- \frac{2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} + \frac{- 200 | x^{2} - 100 | | (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |) - 200 | x^{2} - 100 | | (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8

Simplify the numerator.

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Step 2.5.4.8.1

Factor $2$ out of $- 2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |) - 200 | x^{2} - 100 | | (x + 10) (x - 10) |$ .

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Step 2.5.4.8.1.1

Factor $2$ out of $- 2 x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 (- x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)) - 200 | x^{2} - 100 | | (x + 10) (x - 10) |}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.1.2

Factor $2$ out of $- 200 | x^{2} - 100 | | (x + 10) (x - 10) |$ .

$f'' (x) = \frac{\frac{2 (- x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)) + 2 (- 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.1.3

Factor $2$ out of $2 (- x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |)) + 2 (- 100 | x^{2} - 100 | | (x + 10) (x - 10) |)$ .

$f'' (x) = \frac{\frac{2 (- x^{2} (2 x^{4} - 400 x^{2} + 20000 - 3 | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- x^{2} (2 x^{4}) - x^{2} (- 400 x^{2}) - x^{2} \cdot 20000 - x^{2} (- 3 | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.3

Simplify.

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Step 2.5.4.8.3.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 x^{2} x^{4}) - x^{2} (- 400 x^{2}) - x^{2} \cdot 20000 - x^{2} (- 3 | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.3.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 x^{2} x^{4}) - 1 \cdot (- 400 x^{2} x^{2}) - x^{2} \cdot 20000 - x^{2} (- 3 | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.3.3

Multiply $20000$ by $- 1$ .

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 x^{2} x^{4}) - 1 \cdot (- 400 x^{2} x^{2}) - 20000 x^{2} - x^{2} (- 3 | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.3.4

Multiply $- 3$ by $- 1$ .

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 x^{2} x^{4}) - 1 \cdot (- 400 x^{2} x^{2}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4

Simplify each term.

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Step 2.5.4.8.4.1

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

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Step 2.5.4.8.4.1.1

Move $x^{4}$ .

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 (x^{4} x^{2})) - 1 \cdot (- 400 x^{2} x^{2}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.1.2

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

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 x^{4 + 2}) - 1 \cdot (- 400 x^{2} x^{2}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.1.3

Add $4$ and $2$ .

$f'' (x) = \frac{\frac{2 (- 1 \cdot (2 x^{6}) - 1 \cdot (- 400 x^{2} x^{2}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.2

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} - 1 \cdot (- 400 x^{2} x^{2}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.3

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

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Step 2.5.4.8.4.3.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} - 1 \cdot (- 400 (x^{2} x^{2})) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.3.2

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

$f'' (x) = \frac{\frac{2 (- 2 x^{6} - 1 \cdot (- 400 x^{2 + 2}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.3.3

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} - 1 \cdot (- 400 x^{4}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.4.4

Multiply $- 1$ by $- 400$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | (x + 10) (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.5

Expand $(x + 10) (x - 10)$ using the FOIL Method.

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Step 2.5.4.8.5.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x (x - 10) + 10 (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.5.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x \cdot x + x \cdot - 10 + 10 (x - 10) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.5.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.6

Combine the opposite terms in $x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10$ .

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Step 2.5.4.8.6.1

Reorder the factors in the terms $x \cdot - 10$ and $10 x$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x \cdot x - 10 x + 10 x + 10 \cdot - 10 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.6.2

Add $- 10 x$ and $10 x$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x \cdot x + 0 + 10 \cdot - 10 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.6.3

Add $x \cdot x$ and $0$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x \cdot x + 10 \cdot - 10 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.7

Simplify each term.

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Step 2.5.4.8.7.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x^{2} + 10 \cdot - 10 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.7.2

Multiply $10$ by $- 10$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} - 100 | | x^{2} - 100 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.8

Multiply $- 100 | x^{2} - 100 | | x^{2} - 100 |$ .

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Step 2.5.4.8.8.1

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

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | (x^{2} - 100) (x^{2} - 100) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.8.2

Raise $x^{2} - 100$ to the power of $1$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | (x^{2} - 100) (x^{2} - 100) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.8.3

Raise $x^{2} - 100$ to the power of $1$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | (x^{2} - 100) (x^{2} - 100) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.8.4

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

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | {(x^{2} - 100)}^{1 + 1} |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.8.5

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | {(x^{2} - 100)}^{2} |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.9

Rewrite ${(x^{2} - 100)}^{2}$ as $(x^{2} - 100) (x^{2} - 100)$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | (x^{2} - 100) (x^{2} - 100) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.10

Expand $(x^{2} - 100) (x^{2} - 100)$ using the FOIL Method.

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Step 2.5.4.8.10.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} (x^{2} - 100) - 100 (x^{2} - 100) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.10.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} x^{2} + x^{2} \cdot - 100 - 100 (x^{2} - 100) |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.10.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2} x^{2} + x^{2} \cdot - 100 - 100 x^{2} - 100 \cdot - 100 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.11

Simplify and combine like terms.

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Step 2.5.4.8.11.1

Simplify each term.

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Step 2.5.4.8.11.1.1

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

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Step 2.5.4.8.11.1.1.1

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

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{2 + 2} + x^{2} \cdot - 100 - 100 x^{2} - 100 \cdot - 100 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.11.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} + x^{2} \cdot - 100 - 100 x^{2} - 100 \cdot - 100 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.11.1.2

Move $- 100$ to the left of $x^{2}$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 100 \cdot x^{2} - 100 x^{2} - 100 \cdot - 100 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.11.1.3

Multiply $- 100$ by $- 100$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 100 x^{2} - 100 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.8.11.2

Subtract $100 x^{2}$ from $- 100 x^{2}$ .

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.9

Factor $- 1$ out of $- 2 x^{6}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6}) + 400 x^{4} - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.10

Factor $- 1$ out of $400 x^{4}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6}) - (- 400 x^{4}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.11

Factor $- 1$ out of $- (2 x^{6}) - (- 400 x^{4})$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4}) - 20000 x^{2} + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.12

Factor $- 1$ out of $- 20000 x^{2}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4}) - (20000 x^{2}) + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.13

Factor $- 1$ out of $- (2 x^{6} - 400 x^{4}) - (20000 x^{2})$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4} + 20000 x^{2}) + 3 x^{2} | x^{4} - 200 x^{2} + 10000 | - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.14

Factor $- 1$ out of $3 x^{2} | x^{4} - 200 x^{2} + 10000 |$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4} + 20000 x^{2}) - (- 3 x^{2} | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.15

Factor $- 1$ out of $- (2 x^{6} - 400 x^{4} + 20000 x^{2}) - (- 3 x^{2} | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 |) - 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.16

Factor $- 1$ out of $- 100 | x^{4} - 200 x^{2} + 10000 |$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 |) - (100 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.17

Factor $- 1$ out of $- (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 |) - (100 | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.18

Rewrite $- (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)$ as $- 1 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)$ .

$f'' (x) = \frac{\frac{2 (- 1 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |))}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.4.19

Move the negative in front of the fraction.

$f'' (x) = \frac{- \frac{2 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$

Step 2.5.5

Combine terms.

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Step 2.5.5.1

Rewrite $\frac{- \frac{2 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}}{{| x^{2} - 100 |}^{2}}$ as a product.

$f'' (x) = - \frac{2 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |} \cdot \frac{1}{{| x^{2} - 100 |}^{2}}$

Step 2.5.5.2

Multiply $\frac{1}{{| x^{2} - 100 |}^{2}}$ by $\frac{2 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)}{| (x + 10) (x - 10) |}$ .

$f'' (x) = - \frac{2 (2 x^{6} - 400 x^{4} + 20000 x^{2} - 3 x^{2} | x^{4} - 200 x^{2} + 10000 | + 100 | x^{4} - 200 x^{2} + 10000 |)}{{| x^{2} - 100 |}^{2} | (x + 10) (x - 10) |}$

Step 3

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

$\frac{2 x^{3} - 200 x}{| x^{2} - 100 |} = 0$

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 $g (x) = x^{2} - 100$ .

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Step 4.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 100$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - 100]$

Step 4.1.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [x^{2} - 100]$

Step 4.1.1.3

Replace all occurrences of $u$ with $x^{2} - 100$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} \frac{d}{d x} [x^{2} - 100]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $x^{2} - 100$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 100]$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 100])$

Step 4.1.2.2

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

$\frac{x^{2} - 100}{| x^{2} - 100 |} (2 x + \frac{d}{d x} [- 100])$

Step 4.1.2.3

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100$ with respect to $x$ is $0$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} (2 x + 0)$

Step 4.1.2.4

Combine fractions.

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Step 4.1.2.4.1

Add $2 x$ and $0$ .

$\frac{x^{2} - 100}{| x^{2} - 100 |} (2 x)$

Step 4.1.2.4.2

Combine $2$ and $\frac{x^{2} - 100}{| x^{2} - 100 |}$ .

$\frac{2 (x^{2} - 100)}{| x^{2} - 100 |} x$

Step 4.1.2.4.3

Combine $\frac{2 (x^{2} - 100)}{| x^{2} - 100 |}$ and $x$ .

$\frac{2 (x^{2} - 100) x}{| x^{2} - 100 |}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot - 100) x}{| x^{2} - 100 |}$

Step 4.1.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 4.1.3.3

Simplify each term.

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Step 4.1.3.3.1

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

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Step 4.1.3.3.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 4.1.3.3.1.2

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

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Step 4.1.3.3.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 4.1.3.3.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 4.1.3.3.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot - 100 x}{| x^{2} - 100 |}$

Step 4.1.3.3.2

Multiply $2$ by $- 100$ .

$f' (x) = \frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$ .

$\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{2 x^{3} - 200 x}{| x^{2} - 100 |} = 0$

Step 5.2

Set the numerator equal to zero.

$2 x^{3} - 200 x = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Factor the left side of the equation.

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Step 5.3.1.1

Factor $2 x$ out of $2 x^{3} - 200 x$ .

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Step 5.3.1.1.1

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

$2 x (x^{2}) - 200 x = 0$

Step 5.3.1.1.2

Factor $2 x$ out of $- 200 x$ .

$2 x (x^{2}) + 2 x (- 100) = 0$

Step 5.3.1.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 100)$ .

$2 x (x^{2} - 100) = 0$

Step 5.3.1.2

Rewrite $100$ as $10^{2}$ .

$2 x (x^{2} - 10^{2}) = 0$

Step 5.3.1.3

Factor.

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Step 5.3.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 10$ .

$2 x ((x + 10) (x - 10)) = 0$

Step 5.3.1.3.2

Remove unnecessary parentheses.

$2 x (x + 10) (x - 10) = 0$

Step 5.3.2

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

$x = 0$

$x + 10 = 0$

$x - 10 = 0$

Step 5.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.4

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 5.3.4.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 5.3.4.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 5.3.5

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 5.3.5.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 5.3.5.2

Add $10$ to both sides of the equation.

$x = 10$

Step 5.3.6

The final solution is all the values that make $2 x (x + 10) (x - 10) = 0$ true.

$x = 0, - 10, 10$

Step 5.4

Exclude the solutions that do not make $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |} = 0$ true.

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$ equal to $0$ to find where the expression is undefined.

$| x^{2} - 100 | = 0$

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 $| x | = \pm x$ .

$x^{2} - 100 = \pm 0$

Step 6.2.2

Plus or minus $0$ is $0$ .

$x^{2} - 100 = 0$

Step 6.2.3

Add $100$ to both sides of the equation.

$x^{2} = 100$

Step 6.2.4

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

$x = \pm \sqrt{100}$

Step 6.2.5

Simplify $\pm \sqrt{100}$ .

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Step 6.2.5.1

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2}}$

Step 6.2.5.2

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

$x = \pm 10$

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.

$x = 10$

Step 6.2.6.2

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

$x = - 10$

Step 6.2.6.3

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

$x = 10, - 10$

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 $0$ .

$x = - 10, x = 10$

Step 7

Critical points to evaluate.

$x = 0, - 10, 10$

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.

$- \frac{2 (2 {(0)}^{6} - 400 {(0)}^{4} + 20000 {(0)}^{2} - 3 {(0)}^{2} | {(0)}^{4} - 200 {(0)}^{2} + 10000 | + 100 | {(0)}^{4} - 200 {(0)}^{2} + 10000 |)}{{| {(0)}^{2} - 100 |}^{2} | ((0) + 10) ((0) - 10) |}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$- \frac{2 (2 \cdot 0 - 400 \cdot 0^{4} + 20000 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.2

Multiply $2$ by $0$ .

$- \frac{2 (0 - 400 \cdot 0^{4} + 20000 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.3

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

$- \frac{2 (0 - 400 \cdot 0 + 20000 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.4

Multiply $- 400$ by $0$ .

$- \frac{2 (0 + 0 + 20000 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.5

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

$- \frac{2 (0 + 0 + 20000 \cdot 0 - 3 \cdot 0^{2} | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.6

Multiply $20000$ by $0$ .

$- \frac{2 (0 + 0 + 0 - 3 \cdot 0^{2} | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.7

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

$- \frac{2 (0 + 0 + 0 - 3 \cdot 0 | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.8

Multiply $- 3$ by $0$ .

$- \frac{2 (0 + 0 + 0 + 0 | 0^{4} - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.9

Simplify each term.

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Step 9.1.9.1

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

$- \frac{2 (0 + 0 + 0 + 0 | 0 - 200 \cdot 0^{2} + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.9.2

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

$- \frac{2 (0 + 0 + 0 + 0 | 0 - 200 \cdot 0 + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.9.3

Multiply $- 200$ by $0$ .

$- \frac{2 (0 + 0 + 0 + 0 | 0 + 0 + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.10

Add $0$ and $0$ .

$- \frac{2 (0 + 0 + 0 + 0 | 0 + 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.11

Add $0$ and $10000$ .

$- \frac{2 (0 + 0 + 0 + 0 | 10000 | + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.12

The absolute value is the distance between a number and zero. The distance between $0$ and $10000$ is $10000$ .

$- \frac{2 (0 + 0 + 0 + 0 \cdot 10000 + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.13

Multiply $0$ by $10000$ .

$- \frac{2 (0 + 0 + 0 + 0 + 100 | 0^{4} - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.14

Simplify each term.

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Step 9.1.14.1

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

$- \frac{2 (0 + 0 + 0 + 0 + 100 | 0 - 200 \cdot 0^{2} + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.14.2

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

$- \frac{2 (0 + 0 + 0 + 0 + 100 | 0 - 200 \cdot 0 + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.14.3

Multiply $- 200$ by $0$ .

$- \frac{2 (0 + 0 + 0 + 0 + 100 | 0 + 0 + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.15

Add $0$ and $0$ .

$- \frac{2 (0 + 0 + 0 + 0 + 100 | 0 + 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.16

Add $0$ and $10000$ .

$- \frac{2 (0 + 0 + 0 + 0 + 100 | 10000 |)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.17

The absolute value is the distance between a number and zero. The distance between $0$ and $10000$ is $10000$ .

$- \frac{2 (0 + 0 + 0 + 0 + 100 \cdot 10000)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.18

Multiply $100$ by $10000$ .

$- \frac{2 (0 + 0 + 0 + 0 + 1000000)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.19

Add $0$ and $0$ .

$- \frac{2 (0 + 0 + 0 + 1000000)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.20

Add $0$ and $0$ .

$- \frac{2 (0 + 0 + 1000000)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.21

Add $0$ and $0$ .

$- \frac{2 (0 + 1000000)}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.1.22

Add $0$ and $1000000$ .

$- \frac{2 \cdot 1000000}{{| 0^{2} - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$- \frac{2 \cdot 1000000}{{| 0 - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.2.2

Subtract $100$ from $0$ .

$- \frac{2 \cdot 1000000}{{| - 100 |}^{2} | (0 + 10) (0 - 10) |}$

Step 9.2.3

Add $0$ and $10$ .

$- \frac{2 \cdot 1000000}{{| - 100 |}^{2} | 10 (0 - 10) |}$

Step 9.2.4

Subtract $10$ from $0$ .

$- \frac{2 \cdot 1000000}{{| - 100 |}^{2} | 10 \cdot - 10 |}$

Step 9.2.5

The absolute value is the distance between a number and zero. The distance between $- 100$ and $0$ is $100$ .

$- \frac{2 \cdot 1000000}{100^{2} | 10 \cdot - 10 |}$

Step 9.2.6

Raise $100$ to the power of $2$ .

$- \frac{2 \cdot 1000000}{10000 | 10 \cdot - 10 |}$

Step 9.2.7

Multiply $10$ by $- 10$ .

$- \frac{2 \cdot 1000000}{10000 | - 100 |}$

Step 9.2.8

The absolute value is the distance between a number and zero. The distance between $- 100$ and $0$ is $100$ .

$- \frac{2 \cdot 1000000}{10000 \cdot 100}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Multiply $2$ by $1000000$ .

$- \frac{2000000}{10000 \cdot 100}$

Step 9.3.2

Multiply $10000$ by $100$ .

$- \frac{2000000}{1000000}$

Step 9.3.3

Divide $2000000$ by $1000000$ .

$- 1 \cdot 2$

Step 9.3.4

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.

$f (0) = | {(0)}^{2} - 100 |$

Step 11.2

Simplify the result.

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Step 11.2.1

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

$f (0) = | 0 - 100 |$

Step 11.2.2

Subtract $100$ from $0$ .

$f (0) = | - 100 |$

Step 11.2.3

The absolute value is the distance between a number and zero. The distance between $- 100$ and $0$ is $100$ .

$f (0) = 100$

Step 11.2.4

The final answer is $100$ .

$y = 100$

Step 12

Evaluate the second derivative at $x = - 10$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{{| {(- 10)}^{2} - 100 |}^{2} | ((- 10) + 10) ((- 10) - 10) |}$

Step 13

Evaluate the second derivative.

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Step 13.1

Raise $- 10$ to the power of $2$ .

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{{| 100 - 100 |}^{2} | ((- 10) + 10) ((- 10) - 10) |}$

Step 13.2

Subtract $100$ from $100$ .

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{{| 0 |}^{2} | ((- 10) + 10) ((- 10) - 10) |}$

Step 13.3

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

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0^{2} | ((- 10) + 10) ((- 10) - 10) |}$

Step 13.4

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

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0 | ((- 10) + 10) ((- 10) - 10) |}$

Step 13.5

Add $- 10$ and $10$ .

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0 | 0 ((- 10) - 10) |}$

Step 13.6

Subtract $10$ from $- 10$ .

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0 | 0 \cdot - 20 |}$

Step 13.7

Multiply $0$ by $- 20$ .

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0 | 0 |}$

Step 13.8

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

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0 \cdot 0}$

Step 13.9

Multiply $0$ by $0$ .

$- \frac{2 (2 {(- 10)}^{6} - 400 {(- 10)}^{4} + 20000 {(- 10)}^{2} - 3 {(- 10)}^{2} | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 | + 100 | {(- 10)}^{4} - 200 {(- 10)}^{2} + 10000 |)}{0}$

Step 13.10

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.

$(- \infty, - 10) \cup (- 10, 0) \cup (0, 10) \cup (10, \infty)$

Step 14.2

Substitute any number, such as $- 12$ , from the interval $(- \infty, - 10)$ in the first derivative $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 12$ in the expression.

$f' (- 12) = \frac{2 {(- 12)}^{3} - 200 \cdot - 12}{| {(- 12)}^{2} - 100 |}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify the numerator.

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Step 14.2.2.1.1

Raise $- 12$ to the power of $3$ .

$f' (- 12) = \frac{2 \cdot - 1728 - 200 \cdot - 12}{| {(- 12)}^{2} - 100 |}$

Step 14.2.2.1.2

Multiply $2$ by $- 1728$ .

$f' (- 12) = \frac{- 3456 - 200 \cdot - 12}{| {(- 12)}^{2} - 100 |}$

Step 14.2.2.1.3

Multiply $- 200$ by $- 12$ .

$f' (- 12) = \frac{- 3456 + 2400}{| {(- 12)}^{2} - 100 |}$

Step 14.2.2.1.4

Add $- 3456$ and $2400$ .

$f' (- 12) = \frac{- 1056}{| {(- 12)}^{2} - 100 |}$

Step 14.2.2.2

Simplify the denominator.

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Step 14.2.2.2.1

Raise $- 12$ to the power of $2$ .

$f' (- 12) = \frac{- 1056}{| 144 - 100 |}$

Step 14.2.2.2.2

Subtract $100$ from $144$ .

$f' (- 12) = \frac{- 1056}{| 44 |}$

Step 14.2.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $44$ is $44$ .

$f' (- 12) = \frac{- 1056}{44}$

Step 14.2.2.3

Divide $- 1056$ by $44$ .

$f' (- 12) = - 24$

Step 14.2.2.4

The final answer is $- 24$ .

$- 24$

Step 14.3

Substitute any number, such as $- 5$ , from the interval $(- 10, 0)$ in the first derivative $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$ to check if the result is negative or positive.

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Step 14.3.1

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

$f' (- 5) = \frac{2 {(- 5)}^{3} - 200 \cdot - 5}{| {(- 5)}^{2} - 100 |}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify the numerator.

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Step 14.3.2.1.1

Raise $- 5$ to the power of $3$ .

$f' (- 5) = \frac{2 \cdot - 125 - 200 \cdot - 5}{| {(- 5)}^{2} - 100 |}$

Step 14.3.2.1.2

Multiply $2$ by $- 125$ .

$f' (- 5) = \frac{- 250 - 200 \cdot - 5}{| {(- 5)}^{2} - 100 |}$

Step 14.3.2.1.3

Multiply $- 200$ by $- 5$ .

$f' (- 5) = \frac{- 250 + 1000}{| {(- 5)}^{2} - 100 |}$

Step 14.3.2.1.4

Add $- 250$ and $1000$ .

$f' (- 5) = \frac{750}{| {(- 5)}^{2} - 100 |}$

Step 14.3.2.2

Simplify the denominator.

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Step 14.3.2.2.1

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

$f' (- 5) = \frac{750}{| 25 - 100 |}$

Step 14.3.2.2.2

Subtract $100$ from $25$ .

$f' (- 5) = \frac{750}{| - 75 |}$

Step 14.3.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 75$ and $0$ is $75$ .

$f' (- 5) = \frac{750}{75}$

Step 14.3.2.3

Divide $750$ by $75$ .

$f' (- 5) = 10$

Step 14.3.2.4

The final answer is $10$ .

$10$

Step 14.4

Substitute any number, such as $5$ , from the interval $(0, 10)$ in the first derivative $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$ to check if the result is negative or positive.

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Step 14.4.1

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

$f' (5) = \frac{2 {(5)}^{3} - 200 \cdot 5}{| {(5)}^{2} - 100 |}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify the numerator.

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Step 14.4.2.1.1

Raise $5$ to the power of $3$ .

$f' (5) = \frac{2 \cdot 125 - 200 \cdot 5}{| 5^{2} - 100 |}$

Step 14.4.2.1.2

Multiply $2$ by $125$ .

$f' (5) = \frac{250 - 200 \cdot 5}{| 5^{2} - 100 |}$

Step 14.4.2.1.3

Multiply $- 200$ by $5$ .

$f' (5) = \frac{250 - 1000}{| 5^{2} - 100 |}$

Step 14.4.2.1.4

Subtract $1000$ from $250$ .

$f' (5) = \frac{- 750}{| 5^{2} - 100 |}$

Step 14.4.2.2

Simplify the denominator.

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Step 14.4.2.2.1

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

$f' (5) = \frac{- 750}{| 25 - 100 |}$

Step 14.4.2.2.2

Subtract $100$ from $25$ .

$f' (5) = \frac{- 750}{| - 75 |}$

Step 14.4.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 75$ and $0$ is $75$ .

$f' (5) = \frac{- 750}{75}$

Step 14.4.2.3

Divide $- 750$ by $75$ .

$f' (5) = - 10$

Step 14.4.2.4

The final answer is $- 10$ .

$- 10$

Step 14.5

Substitute any number, such as $12$ , from the interval $(10, \infty)$ in the first derivative $\frac{2 x^{3} - 200 x}{| x^{2} - 100 |}$ to check if the result is negative or positive.

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Step 14.5.1

Replace the variable $x$ with $12$ in the expression.

$f' (12) = \frac{2 {(12)}^{3} - 200 \cdot 12}{| {(12)}^{2} - 100 |}$

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Simplify the numerator.

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Step 14.5.2.1.1

Raise $12$ to the power of $3$ .

$f' (12) = \frac{2 \cdot 1728 - 200 \cdot 12}{| 12^{2} - 100 |}$

Step 14.5.2.1.2

Multiply $2$ by $1728$ .

$f' (12) = \frac{3456 - 200 \cdot 12}{| 12^{2} - 100 |}$

Step 14.5.2.1.3

Multiply $- 200$ by $12$ .

$f' (12) = \frac{3456 - 2400}{| 12^{2} - 100 |}$

Step 14.5.2.1.4

Subtract $2400$ from $3456$ .

$f' (12) = \frac{1056}{| 12^{2} - 100 |}$

Step 14.5.2.2

Simplify the denominator.

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Step 14.5.2.2.1

Raise $12$ to the power of $2$ .

$f' (12) = \frac{1056}{| 144 - 100 |}$

Step 14.5.2.2.2

Subtract $100$ from $144$ .

$f' (12) = \frac{1056}{| 44 |}$

Step 14.5.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $44$ is $44$ .

$f' (12) = \frac{1056}{44}$

Step 14.5.2.3

Divide $1056$ by $44$ .

$f' (12) = 24$

Step 14.5.2.4

The final answer is $24$ .

$24$

Step 14.6

Since the first derivative changed signs from negative to positive around $x = - 10$ , then $x = - 10$ is a local minimum.

$x = - 10$ 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 $x = 10$ , then $x = 10$ is a local minimum.

$x = 10$ is a local minimum

Step 14.9

These are the local extrema for $f (x) = | x^{2} - 100 |$ .

$x = - 10$ is a local minimum

$x = 0$ is a local maximum

$x = 10$ is a local minimum

$x = - 10$ is a local minimum

$x = 0$ is a local maximum

$x = 10$ is a local minimum

Step 15