Calculus Examples

$f (x) = \frac{8 x}{x^{2} - 36}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 x}{x^{2} - 36}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x}{x^{2} - 36}]$ .

$f' (x) = 8 \frac{d}{d x} (\frac{x}{x^{2} - 36})$

Step 1.1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = x^{2} - 36$ .

$f' (x) = 8 (\frac{(x^{2} - 36) \frac{d}{d x} (x) - x \frac{d}{d x} (x^{2} - 36)}{{(x^{2} - 36)}^{2}})$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$f' (x) = 8 (\frac{(x^{2} - 36) \cdot 1 - x \frac{d}{d x} (x^{2} - 36)}{{(x^{2} - 36)}^{2}})$

Step 1.1.3.2

Multiply $x^{2} - 36$ by $1$ .

$f' (x) = 8 (\frac{x^{2} - 36 - x \frac{d}{d x} (x^{2} - 36)}{{(x^{2} - 36)}^{2}})$

Step 1.1.3.3

By the Sum Rule, the derivative of $x^{2} - 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]$ .

$f' (x) = 8 (\frac{x^{2} - 36 - x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 36))}{{(x^{2} - 36)}^{2}})$

Step 1.1.3.4

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

$f' (x) = 8 (\frac{x^{2} - 36 - x (2 x + \frac{d}{d x} (- 36))}{{(x^{2} - 36)}^{2}})$

Step 1.1.3.5

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36$ with respect to $x$ is $0$ .

$f' (x) = 8 (\frac{x^{2} - 36 - x (2 x + 0)}{{(x^{2} - 36)}^{2}})$

Step 1.1.3.6

Simplify the expression.

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Step 1.1.3.6.1

Add $2 x$ and $0$ .

$f' (x) = 8 (\frac{x^{2} - 36 - x (2 x)}{{(x^{2} - 36)}^{2}})$

Step 1.1.3.6.2

Multiply $2$ by $- 1$ .

$f' (x) = 8 (\frac{x^{2} - 36 - 2 x \cdot x}{{(x^{2} - 36)}^{2}})$

Step 1.1.4

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

$f' (x) = 8 (\frac{x^{2} - 36 - 2 (x \cdot x)}{{(x^{2} - 36)}^{2}})$

Step 1.1.5

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

$f' (x) = 8 (\frac{x^{2} - 36 - 2 (x \cdot x)}{{(x^{2} - 36)}^{2}})$

Step 1.1.6

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

$f' (x) = 8 (\frac{x^{2} - 36 - 2 x^{1 + 1}}{{(x^{2} - 36)}^{2}})$

Step 1.1.7

Add $1$ and $1$ .

$f' (x) = 8 (\frac{x^{2} - 36 - 2 x^{2}}{{(x^{2} - 36)}^{2}})$

Step 1.1.8

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

$f' (x) = 8 (\frac{- x^{2} - 36}{{(x^{2} - 36)}^{2}})$

Step 1.1.9

Combine $8$ and $\frac{- x^{2} - 36}{{(x^{2} - 36)}^{2}}$ .

$f' (x) = \frac{8 (- x^{2} - 36)}{{(x^{2} - 36)}^{2}}$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Apply the distributive property.

$f' (x) = \frac{8 (- x^{2}) + 8 \cdot - 36}{{(x^{2} - 36)}^{2}}$

Step 1.1.10.2

Simplify each term.

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Step 1.1.10.2.1

Multiply $- 1$ by $8$ .

$f' (x) = \frac{- 8 x^{2} + 8 \cdot - 36}{{(x^{2} - 36)}^{2}}$

Step 1.1.10.2.2

Multiply $8$ by $- 36$ .

$f' (x) = \frac{- 8 x^{2} - 288}{{(x^{2} - 36)}^{2}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

$f'' (x) = \frac{{(x^{2} - 36)}^{2} \frac{d}{d x} (- 8 x^{2} - 288) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{({(x^{2} - 36)}^{2})}^{2}}$

Step 1.2.2

Differentiate.

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Step 1.2.2.1

Multiply the exponents in ${({(x^{2} - 36)}^{2})}^{2}$ .

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Step 1.2.2.1.1

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

$f'' (x) = \frac{{(x^{2} - 36)}^{2} \frac{d}{d x} (- 8 x^{2} - 288) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{2 \cdot 2}}$

Step 1.2.2.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} \frac{d}{d x} (- 8 x^{2} - 288) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.2.2

By the Sum Rule, the derivative of $- 8 x^{2} - 288$ with respect to $x$ is $\frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 288]$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (\frac{d}{d x} (- 8 x^{2}) + \frac{d}{d x} (- 288)) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 8 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 288)) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.2.4

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

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 8 (2 x) + \frac{d}{d x} (- 288)) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.2.5

Multiply $2$ by $- 8$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x + \frac{d}{d x} (- 288)) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.2.6

Since $- 288$ is constant with respect to $x$ , the derivative of $- 288$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x + 0) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.2.7

Add $- 16 x$ and $0$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - (- 8 x^{2} - 288) \frac{d}{d x} {(x^{2} - 36)}^{2}}{{(x^{2} - 36)}^{4}}$

Step 1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 36$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 36$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - (- 8 x^{2} - 288) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} - 36))}{{(x^{2} - 36)}^{4}}$

Step 1.2.3.2

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

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - (- 8 x^{2} - 288) (2 u \frac{d}{d x} (x^{2} - 36))}{{(x^{2} - 36)}^{4}}$

Step 1.2.3.3

Replace all occurrences of $u$ with $x^{2} - 36$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - (- 8 x^{2} - 288) (2 (x^{2} - 36) \frac{d}{d x} (x^{2} - 36))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4

Differentiate.

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Step 1.2.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 2 (- 8 x^{2} - 288) ((x^{2} - 36) \frac{d}{d x} (x^{2} - 36))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4.2

By the Sum Rule, the derivative of $x^{2} - 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 2 (- 8 x^{2} - 288) ((x^{2} - 36) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 36)))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4.3

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

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 2 (- 8 x^{2} - 288) ((x^{2} - 36) (2 x + \frac{d}{d x} (- 36)))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4.4

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 2 (- 8 x^{2} - 288) ((x^{2} - 36) (2 x + 0))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4.5

Simplify the expression.

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Step 1.2.4.5.1

Add $2 x$ and $0$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 2 (- 8 x^{2} - 288) ((x^{2} - 36) (2 x))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4.5.2

Move $2$ to the left of $x^{2} - 36$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 2 (- 8 x^{2} - 288) (2 \cdot ((x^{2} - 36) x))}{{(x^{2} - 36)}^{4}}$

Step 1.2.4.5.3

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) - 4 (- 8 x^{2} - 288) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Apply the distributive property.

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) + (- 4 (- 8 x^{2}) - 4 \cdot - 288) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.2

Apply the distributive property.

$f'' (x) = \frac{{(x^{2} - 36)}^{2} (- 16 x) + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3

Simplify the numerator.

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Step 1.2.5.3.1

Simplify each term.

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Step 1.2.5.3.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 16 {(x^{2} - 36)}^{2} x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.2

Rewrite ${(x^{2} - 36)}^{2}$ as $(x^{2} - 36) (x^{2} - 36)$ .

$f'' (x) = \frac{- 16 ((x^{2} - 36) (x^{2} - 36)) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.3

Expand $(x^{2} - 36) (x^{2} - 36)$ using the FOIL Method.

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Step 1.2.5.3.1.3.1

Apply the distributive property.

$f'' (x) = \frac{- 16 (x^{2} (x^{2} - 36) - 36 (x^{2} - 36)) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.3.2

Apply the distributive property.

$f'' (x) = \frac{- 16 (x^{2} x^{2} + x^{2} \cdot - 36 - 36 (x^{2} - 36)) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.3.3

Apply the distributive property.

$f'' (x) = \frac{- 16 (x^{2} x^{2} + x^{2} \cdot - 36 - 36 x^{2} - 36 \cdot - 36) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.4

Simplify and combine like terms.

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Step 1.2.5.3.1.4.1

Simplify each term.

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Step 1.2.5.3.1.4.1.1

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

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Step 1.2.5.3.1.4.1.1.1

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

$f'' (x) = \frac{- 16 (x^{2 + 2} + x^{2} \cdot - 36 - 36 x^{2} - 36 \cdot - 36) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.4.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{- 16 (x^{4} + x^{2} \cdot - 36 - 36 x^{2} - 36 \cdot - 36) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.4.1.2

Move $- 36$ to the left of $x^{2}$ .

$f'' (x) = \frac{- 16 (x^{4} - 36 \cdot x^{2} - 36 x^{2} - 36 \cdot - 36) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.4.1.3

Multiply $- 36$ by $- 36$ .

$f'' (x) = \frac{- 16 (x^{4} - 36 x^{2} - 36 x^{2} + 1296) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.4.2

Subtract $36 x^{2}$ from $- 36 x^{2}$ .

$f'' (x) = \frac{- 16 (x^{4} - 72 x^{2} + 1296) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.5

Apply the distributive property.

$f'' (x) = \frac{(- 16 x^{4} - 16 (- 72 x^{2}) - 16 \cdot 1296) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.6

Simplify.

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Step 1.2.5.3.1.6.1

Multiply $- 72$ by $- 16$ .

$f'' (x) = \frac{(- 16 x^{4} + 1152 x^{2} - 16 \cdot 1296) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.6.2

Multiply $- 16$ by $1296$ .

$f'' (x) = \frac{(- 16 x^{4} + 1152 x^{2} - 20736) x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.7

Apply the distributive property.

$f'' (x) = \frac{- 16 x^{4} x + 1152 x^{2} x - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8

Simplify.

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Step 1.2.5.3.1.8.1

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

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Step 1.2.5.3.1.8.1.1

Move $x$ .

$f'' (x) = \frac{- 16 (x \cdot x^{4}) + 1152 x^{2} x - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.1.2

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

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Step 1.2.5.3.1.8.1.2.1

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

$f'' (x) = \frac{- 16 (x \cdot x^{4}) + 1152 x^{2} x - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.1.2.2

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

$f'' (x) = \frac{- 16 x^{1 + 4} + 1152 x^{2} x - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.1.3

Add $1$ and $4$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{2} x - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.2

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

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Step 1.2.5.3.1.8.2.1

Move $x$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 (x \cdot x^{2}) - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.2.2

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

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Step 1.2.5.3.1.8.2.2.1

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

$f'' (x) = \frac{- 16 x^{5} + 1152 (x \cdot x^{2}) - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.2.2.2

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

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{1 + 2} - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.8.2.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + (- 4 (- 8 x^{2}) - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.9

Simplify each term.

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Step 1.2.5.3.1.9.1

Multiply $- 8$ by $- 4$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + (32 x^{2} - 4 \cdot - 288) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.9.2

Multiply $- 4$ by $- 288$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + (32 x^{2} + 1152) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.10

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

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Step 1.2.5.3.1.10.1

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

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Step 1.2.5.3.1.10.1.1

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

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + (32 x^{2} + 1152) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.10.1.2

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

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + (32 x^{2} + 1152) (x^{2 + 1} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.10.2

Add $2$ and $1$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + (32 x^{2} + 1152) (x^{3} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.11

Expand $(32 x^{2} + 1152) (x^{3} - 36 x)$ using the FOIL Method.

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Step 1.2.5.3.1.11.1

Apply the distributive property.

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{2} (x^{3} - 36 x) + 1152 (x^{3} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.11.2

Apply the distributive property.

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{2} x^{3} + 32 x^{2} (- 36 x) + 1152 (x^{3} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.11.3

Apply the distributive property.

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{2} x^{3} + 32 x^{2} (- 36 x) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12

Simplify and combine like terms.

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Step 1.2.5.3.1.12.1

Simplify each term.

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Step 1.2.5.3.1.12.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.2.5.3.1.12.1.1.1

Move $x^{3}$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 (x^{3} x^{2}) + 32 x^{2} (- 36 x) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.1.2

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

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{3 + 2} + 32 x^{2} (- 36 x) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.1.3

Add $3$ and $2$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 32 x^{2} (- 36 x) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 32 \cdot (- 36 x^{2} x) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.3

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

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Step 1.2.5.3.1.12.1.3.1

Move $x$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 32 \cdot (- 36 (x \cdot x^{2})) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.3.2

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

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Step 1.2.5.3.1.12.1.3.2.1

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

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 32 \cdot (- 36 (x \cdot x^{2})) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.3.2.2

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

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 32 \cdot (- 36 x^{1 + 2}) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 32 \cdot (- 36 x^{3}) + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.4

Multiply $32$ by $- 36$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} - 1152 x^{3} + 1152 x^{3} + 1152 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.1.5

Multiply $- 36$ by $1152$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} - 1152 x^{3} + 1152 x^{3} - 41472 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.2

Add $- 1152 x^{3}$ and $1152 x^{3}$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} + 0 - 41472 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.1.12.3

Add $32 x^{5}$ and $0$ .

$f'' (x) = \frac{- 16 x^{5} + 1152 x^{3} - 20736 x + 32 x^{5} - 41472 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.2

Add $- 16 x^{5}$ and $32 x^{5}$ .

$f'' (x) = \frac{16 x^{5} + 1152 x^{3} - 20736 x - 41472 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.3.3

Subtract $41472 x$ from $- 20736 x$ .

$f'' (x) = \frac{16 x^{5} + 1152 x^{3} - 62208 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4

Simplify the numerator.

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Step 1.2.5.4.1

Factor $16 x$ out of $16 x^{5} + 1152 x^{3} - 62208 x$ .

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Step 1.2.5.4.1.1

Factor $16 x$ out of $16 x^{5}$ .

$f'' (x) = \frac{16 x (x^{4}) + 1152 x^{3} - 62208 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.1.2

Factor $16 x$ out of $1152 x^{3}$ .

$f'' (x) = \frac{16 x (x^{4}) + 16 x (72 x^{2}) - 62208 x}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.1.3

Factor $16 x$ out of $- 62208 x$ .

$f'' (x) = \frac{16 x (x^{4}) + 16 x (72 x^{2}) + 16 x (- 3888)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.1.4

Factor $16 x$ out of $16 x (x^{4}) + 16 x (72 x^{2})$ .

$f'' (x) = \frac{16 x (x^{4} + 72 x^{2}) + 16 x (- 3888)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.1.5

Factor $16 x$ out of $16 x (x^{4} + 72 x^{2}) + 16 x (- 3888)$ .

$f'' (x) = \frac{16 x (x^{4} + 72 x^{2} - 3888)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = \frac{16 x ({(x^{2})}^{2} + 72 x^{2} - 3888)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$f'' (x) = \frac{16 x (u^{2} + 72 u - 3888)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.4

Factor $u^{2} + 72 u - 3888$ using the AC method.

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Step 1.2.5.4.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3888$ and whose sum is $72$ .

$- 36, 108$

Step 1.2.5.4.4.2

Write the factored form using these integers.

$f'' (x) = \frac{16 x ((u - 36) (u + 108))}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.5

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

$f'' (x) = \frac{16 x ((x^{2} - 36) (x^{2} + 108))}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.6

Rewrite $36$ as $6^{2}$ .

$f'' (x) = \frac{16 x ((x^{2} - 6^{2}) (x^{2} + 108))}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.4.7

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 $b = 6$ .

$f'' (x) = \frac{16 x (x + 6) (x - 6) (x^{2} + 108)}{{(x^{2} - 36)}^{4}}$

Step 1.2.5.5

Simplify the denominator.

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Step 1.2.5.5.1

Rewrite $36$ as $6^{2}$ .

$f'' (x) = \frac{16 x (x + 6) (x - 6) (x^{2} + 108)}{{(x^{2} - 6^{2})}^{4}}$

Step 1.2.5.5.2

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 $b = 6$ .

$f'' (x) = \frac{16 x (x + 6) (x - 6) (x^{2} + 108)}{{((x + 6) (x - 6))}^{4}}$

Step 1.2.5.5.3

Apply the product rule to $(x + 6) (x - 6)$ .

$f'' (x) = \frac{16 x (x + 6) (x - 6) (x^{2} + 108)}{{(x + 6)}^{4} {(x - 6)}^{4}}$

Step 1.2.5.6

Cancel the common factor of $x + 6$ and ${(x + 6)}^{4}$ .

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Step 1.2.5.6.1

Factor $x + 6$ out of $16 x (x + 6) (x - 6) (x^{2} + 108)$ .

$f'' (x) = \frac{(x + 6) ((16 x (x - 6)) (x^{2} + 108))}{{(x + 6)}^{4} {(x - 6)}^{4}}$

Step 1.2.5.6.2

Cancel the common factors.

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Step 1.2.5.6.2.1

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

$f'' (x) = \frac{(x + 6) ((16 x (x - 6)) (x^{2} + 108))}{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{4})}$

Step 1.2.5.6.2.2

Cancel the common factor.

$f'' (x) = \frac{(x + 6) ((16 x (x - 6)) (x^{2} + 108))}{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{4})}$

Step 1.2.5.6.2.3

Rewrite the expression.

$f'' (x) = \frac{(16 x (x - 6)) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 1.2.5.7

Cancel the common factor of $x - 6$ and ${(x - 6)}^{4}$ .

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Step 1.2.5.7.1

Factor $x - 6$ out of $(16 x (x - 6)) (x^{2} + 108)$ .

$f'' (x) = \frac{(x - 6) ((16 x) (x^{2} + 108))}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 1.2.5.7.2

Cancel the common factors.

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Step 1.2.5.7.2.1

Factor $x - 6$ out of ${(x + 6)}^{3} {(x - 6)}^{4}$ .

$f'' (x) = \frac{(x - 6) ((16 x) (x^{2} + 108))}{(x - 6) ({(x + 6)}^{3} {(x - 6)}^{3})}$

Step 1.2.5.7.2.2

Cancel the common factor.

$f'' (x) = \frac{(x - 6) ((16 x) (x^{2} + 108))}{(x - 6) ({(x + 6)}^{3} {(x - 6)}^{3})}$

Step 1.2.5.7.2.3

Rewrite the expression.

$f'' (x) = \frac{(16 x) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

$f'' (x) = \frac{16 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{16 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$ .

$\frac{16 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{16 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{16 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$16 x (x^{2} + 108) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x = 0$

$x^{2} + 108 = 0$

Step 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

Set $x^{2} + 108$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x^{2} + 108$ equal to $0$ .

$x^{2} + 108 = 0$

Step 2.3.3.2

Solve $x^{2} + 108 = 0$ for $x$ .

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Step 2.3.3.2.1

Subtract $108$ from both sides of the equation.

$x^{2} = - 108$

Step 2.3.3.2.2

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

$x = \pm \sqrt{- 108}$

Step 2.3.3.2.3

Simplify $\pm \sqrt{- 108}$ .

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Step 2.3.3.2.3.1

Rewrite $- 108$ as $- 1 (108)$ .

$x = \pm \sqrt{- 1 (108)}$

Step 2.3.3.2.3.2

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{108}$

Step 2.3.3.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{108}$

Step 2.3.3.2.3.4

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 2.3.3.2.3.4.1

Factor $36$ out of $108$ .

$x = \pm i \cdot \sqrt{36 (3)}$

Step 2.3.3.2.3.4.2

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2} \cdot 3}$

Step 2.3.3.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (6 \sqrt{3})$

Step 2.3.3.2.3.6

Move $6$ to the left of $i$ .

$x = \pm 6 i \sqrt{3}$

Step 2.3.3.2.4

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

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Step 2.3.3.2.4.1

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

$x = 6 i \sqrt{3}$

Step 2.3.3.2.4.2

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

$x = - 6 i \sqrt{3}$

Step 2.3.3.2.4.3

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

$x = 6 i \sqrt{3}, - 6 i \sqrt{3}$

Step 2.3.4

The final solution is all the values that make $16 x (x^{2} + 108) = 0$ true.

$x = 0, 6 i \sqrt{3}, - 6 i \sqrt{3}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $0$ in $f (x) = \frac{8 x}{x^{2} - 36}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = \frac{8 (0)}{{(0)}^{2} - 36}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Multiply $8$ by $0$ .

$f (0) = \frac{0}{0^{2} - 36}$

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

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

$f (0) = \frac{0}{0 - 36}$

Step 3.1.2.2.2

Subtract $36$ from $0$ .

$f (0) = \frac{0}{- 36}$

Step 3.1.2.3

Divide $0$ by $- 36$ .

$f (0) = 0$

Step 3.1.2.4

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = \frac{8 x}{x^{2} - 36}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = \frac{16 (- 0.1) ({(- 0.1)}^{2} + 108)}{{((- 0.1) + 6)}^{3} {((- 0.1) - 6)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $16$ by $- 0.1$ .

$f'' (- 0.1) = \frac{- 1.6 ({(- 0.1)}^{2} + 108)}{{(- 0.1 + 6)}^{3} {(- 0.1 - 6)}^{3}}$

Step 5.2.1.2

Multiply $- 1.6$ by ${(- 0.1)}^{2} + 108$ .

$f'' (- 0.1) = \frac{- 172.816}{{(- 0.1 + 6)}^{3} {(- 0.1 - 6)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Add $- 0.1$ and $6$ .

$f'' (- 0.1) = \frac{- 172.816}{{5.9}^{3} {(- 0.1 - 6)}^{3}}$

Step 5.2.2.2

Subtract $6$ from $- 0.1$ .

$f'' (- 0.1) = \frac{- 172.816}{{5.9}^{3} {(- 6.1)}^{3}}$

Step 5.2.2.3

Raise $5.9$ to the power of $3$ .

$f'' (- 0.1) = \frac{- 172.816}{205.379 {(- 6.1)}^{3}}$

Step 5.2.2.4

Raise $- 6.1$ to the power of $3$ .

$f'' (- 0.1) = \frac{- 172.816}{205.379 \cdot - 226.981}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $205.379$ by $- 226.981$ .

$f'' (- 0.1) = \frac{- 172.816}{- 46617.130799}$

Step 5.2.3.2

Divide $- 172.816$ by $- 46617.130799$ .

$f'' (- 0.1) = 0.00370713$

Step 5.2.4

The final answer is $0.00370713$ .

$0.00370713$

Step 5.3

At $- 0.1$ , the second derivative is $0.00370713$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = \frac{16 (0.1) ({(0.1)}^{2} + 108)}{{((0.1) + 6)}^{3} {((0.1) - 6)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $16$ by $0.1$ .

$f'' (0.1) = \frac{1.6 ({0.1}^{2} + 108)}{{(0.1 + 6)}^{3} {(0.1 - 6)}^{3}}$

Step 6.2.1.2

Multiply $1.6$ by ${0.1}^{2} + 108$ .

$f'' (0.1) = \frac{172.816}{{(0.1 + 6)}^{3} {(0.1 - 6)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $0.1$ and $6$ .

$f'' (0.1) = \frac{172.816}{{6.1}^{3} {(0.1 - 6)}^{3}}$

Step 6.2.2.2

Subtract $6$ from $0.1$ .

$f'' (0.1) = \frac{172.816}{{6.1}^{3} {(- 5.9)}^{3}}$

Step 6.2.2.3

Raise $6.1$ to the power of $3$ .

$f'' (0.1) = \frac{172.816}{226.981 {(- 5.9)}^{3}}$

Step 6.2.2.4

Raise $- 5.9$ to the power of $3$ .

$f'' (0.1) = \frac{172.816}{226.981 \cdot - 205.379}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $226.981$ by $- 205.379$ .

$f'' (0.1) = \frac{172.816}{- 46617.130799}$

Step 6.2.3.2

Divide $172.816$ by $- 46617.130799$ .

$f'' (0.1) = - 0.00370713$

Step 6.2.4

The final answer is $- 0.00370713$ .

$- 0.00370713$

Step 6.3

At $0.1$ , the second derivative is $- 0.00370713$ . Since this is negative, the second derivative is decreasing on the interval $(0, \infty)$

Decreasing on $(0, \infty)$ since $f'' (x) < 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(0, 0)$ .

$(0, 0)$

Step 8