Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

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

Step 1.2.4.2

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $2 x$ and $0$ .

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

Step 1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

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

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Step 1.3.3.1.1.1

Move $x$ .

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

Step 1.3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 1.3.3.1.2

Multiply $- 2$ by $9$ .

$\frac{x^{2} - 2 - 2 x^{2} - 18 x}{{(x^{2} - 2)}^{2}}$

Step 1.3.3.2

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

$\frac{- x^{2} - 2 - 18 x}{{(x^{2} - 2)}^{2}}$

Step 1.3.4

Reorder terms.

$\frac{- x^{2} - 18 x - 2}{{(x^{2} - 2)}^{2}}$

Step 1.3.5

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

$\frac{- (x^{2}) - 18 x - 2}{{(x^{2} - 2)}^{2}}$

Step 1.3.6

Factor $- 1$ out of $- 18 x$ .

$\frac{- (x^{2}) - (18 x) - 2}{{(x^{2} - 2)}^{2}}$

Step 1.3.7

Factor $- 1$ out of $- (x^{2}) - (18 x)$ .

$\frac{- (x^{2} + 18 x) - 2}{{(x^{2} - 2)}^{2}}$

Step 1.3.8

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (x^{2} + 18 x) - 1 (2)}{{(x^{2} - 2)}^{2}}$

Step 1.3.9

Factor $- 1$ out of $- (x^{2} + 18 x) - 1 (2)$ .

$\frac{- (x^{2} + 18 x + 2)}{{(x^{2} - 2)}^{2}}$

Step 1.3.10

Rewrite $- (x^{2} + 18 x + 2)$ as $- 1 (x^{2} + 18 x + 2)$ .

$\frac{- 1 (x^{2} + 18 x + 2)}{{(x^{2} - 2)}^{2}}$

Step 1.3.11

Move the negative in front of the fraction.

$f' (x) = - \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}}$ .

$- \frac{d}{d x} [\frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}}] + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.2

$- \frac{{(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} + 18 x + 2] - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{({(x^{2} - 2)}^{2})}^{2}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3

Differentiate.

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Step 2.3.1

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

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Step 2.3.1.1

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

$- \frac{{(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} + 18 x + 2] - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{2 \cdot 2}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.1.2

Multiply $2$ by $2$ .

$- \frac{{(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} + 18 x + 2] - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.2

By the Sum Rule, the derivative of $x^{2} + 18 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [18 x] + \frac{d}{d x} [2]$ .

$- \frac{{(x^{2} - 2)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [18 x] + \frac{d}{d x} [2]) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.3

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

$- \frac{{(x^{2} - 2)}^{2} (2 x + \frac{d}{d x} [18 x] + \frac{d}{d x} [2]) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.4

Since $18$ is constant with respect to $x$ , the derivative of $18 x$ with respect to $x$ is $18 \frac{d}{d x} [x]$ .

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18 \frac{d}{d x} [x] + \frac{d}{d x} [2]) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.5

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

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18 \cdot 1 + \frac{d}{d x} [2]) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.6

Multiply $18$ by $1$ .

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18 + \frac{d}{d x} [2]) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18 + 0) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.3.8

Add $2 x + 18$ and $0$ .

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18) - (x^{2} + 18 x + 2) \frac{d}{d x} [{(x^{2} - 2)}^{2}]}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.4

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

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Step 2.4.1

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

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18) - (x^{2} + 18 x + 2) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.4.2

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

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18) - (x^{2} + 18 x + 2) (2 u \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.4.3

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

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18) - (x^{2} + 18 x + 2) (2 (x^{2} - 2) \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

$- \frac{{(x^{2} - 2)}^{2} (2 x + 18) - 2 (x^{2} + 18 x + 2) ((x^{2} - 2) \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.5.2

Factor $x^{2} - 2$ out of ${(x^{2} - 2)}^{2} (2 x + 18) - 2 (x^{2} + 18 x + 2) ((x^{2} - 2) \frac{d}{d x} [x^{2} - 2])$ .

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Step 2.5.2.1

Factor $x^{2} - 2$ out of ${(x^{2} - 2)}^{2} (2 x + 18)$ .

$- \frac{(x^{2} - 2) ((x^{2} - 2) (2 x + 18)) - 2 (x^{2} + 18 x + 2) ((x^{2} - 2) \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.5.2.2

Factor $x^{2} - 2$ out of $- 2 (x^{2} + 18 x + 2) ((x^{2} - 2) \frac{d}{d x} [x^{2} - 2])$ .

$- \frac{(x^{2} - 2) ((x^{2} - 2) (2 x + 18)) + (x^{2} - 2) (- 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.5.2.3

Factor $x^{2} - 2$ out of $(x^{2} - 2) ((x^{2} - 2) (2 x + 18)) + (x^{2} - 2) (- 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2} - 2]))$ .

$- \frac{(x^{2} - 2) ((x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{4}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$- \frac{(x^{2} - 2) ((x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2} - 2]))}{(x^{2} - 2) {(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.6.2

Cancel the common factor.

$- \frac{(x^{2} - 2) ((x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2} - 2]))}{(x^{2} - 2) {(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.6.3

Rewrite the expression.

$- \frac{(x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.7

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

$- \frac{(x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.8

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

$- \frac{(x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (2 x + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- \frac{(x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (2 x + 0)}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

$- \frac{(x^{2} - 2) (2 x + 18) - 2 (x^{2} + 18 x + 2) (2 x)}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.10.2

Multiply $2$ by $- 2$ .

$- \frac{(x^{2} - 2) (2 x + 18) - 4 (x^{2} + 18 x + 2) x}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \frac{d}{d x} [- 1]$

Step 2.11

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

$- \frac{(x^{2} - 2) (2 x + 18) - 4 (x^{2} + 18 x + 2) x}{{(x^{2} - 2)}^{3}} + \frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}} \cdot 0$

Step 2.12

Simplify the expression.

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Step 2.12.1

Multiply $\frac{x^{2} + 18 x + 2}{{(x^{2} - 2)}^{2}}$ by $0$ .

$- \frac{(x^{2} - 2) (2 x + 18) - 4 (x^{2} + 18 x + 2) x}{{(x^{2} - 2)}^{3}} + 0$

Step 2.12.2

Add $- \frac{(x^{2} - 2) (2 x + 18) - 4 (x^{2} + 18 x + 2) x}{{(x^{2} - 2)}^{3}}$ and $0$ .

$- \frac{(x^{2} - 2) (2 x + 18) - 4 (x^{2} + 18 x + 2) x}{{(x^{2} - 2)}^{3}}$

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

$- \frac{(x^{2} - 2) (2 x + 18) + (- 4 x^{2} - 4 (18 x) - 4 \cdot 2) x}{{(x^{2} - 2)}^{3}}$

Step 2.13.2

Apply the distributive property.

$- \frac{(x^{2} - 2) (2 x + 18) - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3

Simplify the numerator.

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Step 2.13.3.1

Simplify each term.

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Step 2.13.3.1.1

Expand $(x^{2} - 2) (2 x + 18)$ using the FOIL Method.

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Step 2.13.3.1.1.1

Apply the distributive property.

$- \frac{x^{2} (2 x + 18) - 2 (2 x + 18) - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.1.2

Apply the distributive property.

$- \frac{x^{2} (2 x) + x^{2} \cdot 18 - 2 (2 x + 18) - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.1.3

Apply the distributive property.

$- \frac{x^{2} (2 x) + x^{2} \cdot 18 - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2

Simplify each term.

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Step 2.13.3.1.2.1

Rewrite using the commutative property of multiplication.

$- \frac{2 x^{2} x + x^{2} \cdot 18 - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.2

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

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Step 2.13.3.1.2.2.1

Move $x$ .

$- \frac{2 (x \cdot x^{2}) + x^{2} \cdot 18 - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.2.2

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

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Step 2.13.3.1.2.2.2.1

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

$- \frac{2 (x^{1} x^{2}) + x^{2} \cdot 18 - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.2.2.2

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

$- \frac{2 x^{1 + 2} + x^{2} \cdot 18 - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.2.3

Add $1$ and $2$ .

$- \frac{2 x^{3} + x^{2} \cdot 18 - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.3

Move $18$ to the left of $x^{2}$ .

$- \frac{2 x^{3} + 18 \cdot x^{2} - 2 (2 x) - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.4

Multiply $2$ by $- 2$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 2 \cdot 18 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.2.5

Multiply $- 2$ by $18$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{2} x - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.3

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

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Step 2.13.3.1.3.1

Move $x$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 (x \cdot x^{2}) - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.3.2

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

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Step 2.13.3.1.3.2.1

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

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 (x^{1} x^{2}) - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.3.2.2

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

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{1 + 2} - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.3.3

Add $1$ and $2$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{3} - 4 (18 x) x - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.4

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

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Step 2.13.3.1.4.1

Move $x$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{3} - 4 (18 (x \cdot x)) - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.4.2

Multiply $x$ by $x$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{3} - 4 (18 x^{2}) - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.5

Multiply $18$ by $- 4$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{3} - 72 x^{2} - 4 \cdot 2 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.1.6

Multiply $- 4$ by $2$ .

$- \frac{2 x^{3} + 18 x^{2} - 4 x - 36 - 4 x^{3} - 72 x^{2} - 8 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.2

Subtract $4 x^{3}$ from $2 x^{3}$ .

$- \frac{- 2 x^{3} + 18 x^{2} - 4 x - 36 - 72 x^{2} - 8 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.3

Subtract $72 x^{2}$ from $18 x^{2}$ .

$- \frac{- 2 x^{3} - 54 x^{2} - 4 x - 36 - 8 x}{{(x^{2} - 2)}^{3}}$

Step 2.13.3.4

Subtract $8 x$ from $- 4 x$ .

$- \frac{- 2 x^{3} - 54 x^{2} - 12 x - 36}{{(x^{2} - 2)}^{3}}$

Step 2.13.4

Factor $2$ out of $- 2 x^{3} - 54 x^{2} - 12 x - 36$ .

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Step 2.13.4.1

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

$- \frac{2 (- x^{3}) - 54 x^{2} - 12 x - 36}{{(x^{2} - 2)}^{3}}$

Step 2.13.4.2

Factor $2$ out of $- 54 x^{2}$ .

$- \frac{2 (- x^{3}) + 2 (- 27 x^{2}) - 12 x - 36}{{(x^{2} - 2)}^{3}}$

Step 2.13.4.3

Factor $2$ out of $- 12 x$ .

$- \frac{2 (- x^{3}) + 2 (- 27 x^{2}) + 2 (- 6 x) - 36}{{(x^{2} - 2)}^{3}}$

Step 2.13.4.4

Factor $2$ out of $- 36$ .

$- \frac{2 (- x^{3}) + 2 (- 27 x^{2}) + 2 (- 6 x) + 2 (- 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.4.5

Factor $2$ out of $2 (- x^{3}) + 2 (- 27 x^{2})$ .

$- \frac{2 (- x^{3} - 27 x^{2}) + 2 (- 6 x) + 2 (- 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.4.6

Factor $2$ out of $2 (- x^{3} - 27 x^{2}) + 2 (- 6 x)$ .

$- \frac{2 (- x^{3} - 27 x^{2} - 6 x) + 2 (- 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.4.7

Factor $2$ out of $2 (- x^{3} - 27 x^{2} - 6 x) + 2 (- 18)$ .

$- \frac{2 (- x^{3} - 27 x^{2} - 6 x - 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.5

Factor $- 1$ out of $- x^{3}$ .

$- \frac{2 (- (x^{3}) - 27 x^{2} - 6 x - 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.6

Factor $- 1$ out of $- 27 x^{2}$ .

$- \frac{2 (- (x^{3}) - (27 x^{2}) - 6 x - 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.7

Factor $- 1$ out of $- (x^{3}) - (27 x^{2})$ .

$- \frac{2 (- (x^{3} + 27 x^{2}) - 6 x - 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.8

Factor $- 1$ out of $- 6 x$ .

$- \frac{2 (- (x^{3} + 27 x^{2}) - (6 x) - 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.9

Factor $- 1$ out of $- (x^{3} + 27 x^{2}) - (6 x)$ .

$- \frac{2 (- (x^{3} + 27 x^{2} + 6 x) - 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.10

Rewrite $- 18$ as $- 1 (18)$ .

$- \frac{2 (- (x^{3} + 27 x^{2} + 6 x) - 1 (18))}{{(x^{2} - 2)}^{3}}$

Step 2.13.11

Factor $- 1$ out of $- (x^{3} + 27 x^{2} + 6 x) - 1 (18)$ .

$- \frac{2 (- (x^{3} + 27 x^{2} + 6 x + 18))}{{(x^{2} - 2)}^{3}}$

Step 2.13.12

Rewrite $- (x^{3} + 27 x^{2} + 6 x + 18)$ as $- 1 (x^{3} + 27 x^{2} + 6 x + 18)$ .

$- \frac{2 (- 1 (x^{3} + 27 x^{2} + 6 x + 18))}{{(x^{2} - 2)}^{3}}$

Step 2.13.13

Move the negative in front of the fraction.

$- - \frac{2 (x^{3} + 27 x^{2} + 6 x + 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.14

Multiply $- 1$ by $- 1$ .

$1 \frac{2 (x^{3} + 27 x^{2} + 6 x + 18)}{{(x^{2} - 2)}^{3}}$

Step 2.13.15

Multiply $\frac{2 (x^{3} + 27 x^{2} + 6 x + 18)}{{(x^{2} - 2)}^{3}}$ by $1$ .

$f'' (x) = \frac{2 (x^{3} + 27 x^{2} + 6 x + 18)}{{(x^{2} - 2)}^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 (x^{3} + 27 x^{2} + 6 x + 18)}{{(x^{2} - 2)}^{3}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x^{3} + 27 x^{2} + 6 x + 18}{{(x^{2} - 2)}^{3}}]$ .

$2 \frac{d}{d x} [\frac{x^{3} + 27 x^{2} + 6 x + 18}{{(x^{2} - 2)}^{3}}]$

Step 3.2

$2 \frac{{(x^{2} - 2)}^{3} \frac{d}{d x} [x^{3} + 27 x^{2} + 6 x + 18] - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{({(x^{2} - 2)}^{3})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(x^{2} - 2)}^{3})}^{2}$ .

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Step 3.3.1.1

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

$2 \frac{{(x^{2} - 2)}^{3} \frac{d}{d x} [x^{3} + 27 x^{2} + 6 x + 18] - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{3 \cdot 2}}$

Step 3.3.1.2

Multiply $3$ by $2$ .

$2 \frac{{(x^{2} - 2)}^{3} \frac{d}{d x} [x^{3} + 27 x^{2} + 6 x + 18] - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.2

By the Sum Rule, the derivative of $x^{3} + 27 x^{2} + 6 x + 18$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [27 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [18]$ .

$2 \frac{{(x^{2} - 2)}^{3} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [27 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.3

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

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + \frac{d}{d x} [27 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.4

Since $27$ is constant with respect to $x$ , the derivative of $27 x^{2}$ with respect to $x$ is $27 \frac{d}{d x} [x^{2}]$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 27 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.5

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

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 27 (2 x) + \frac{d}{d x} [6 x] + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.6

Multiply $2$ by $27$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.7

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.8

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

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6 \cdot 1 + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.9

Multiply $6$ by $1$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6 + \frac{d}{d x} [18]) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.10

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6 + 0) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.3.11

Add $3 x^{2} + 54 x + 6$ and $0$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6) - (x^{3} + 27 x^{2} + 6 x + 18) \frac{d}{d x} [{(x^{2} - 2)}^{3}]}{{(x^{2} - 2)}^{6}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 2$ .

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Step 3.4.1

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

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6) - (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{6}}$

Step 3.4.2

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

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6) - (x^{3} + 27 x^{2} + 6 x + 18) (3 u^{2} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{6}}$

Step 3.4.3

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

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6) - (x^{3} + 27 x^{2} + 6 x + 18) (3 {(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{6}}$

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $3$ by $- 1$ .

$2 \frac{{(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) ({(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{6}}$

Step 3.5.2

Factor ${(x^{2} - 2)}^{2}$ out of ${(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) ({(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} - 2])$ .

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Step 3.5.2.1

Factor ${(x^{2} - 2)}^{2}$ out of ${(x^{2} - 2)}^{3} (3 x^{2} + 54 x + 6)$ .

$2 \frac{{(x^{2} - 2)}^{2} ((x^{2} - 2) (3 x^{2} + 54 x + 6)) - 3 (x^{3} + 27 x^{2} + 6 x + 18) ({(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{6}}$

Step 3.5.2.2

Factor ${(x^{2} - 2)}^{2}$ out of $- 3 (x^{3} + 27 x^{2} + 6 x + 18) ({(x^{2} - 2)}^{2} \frac{d}{d x} [x^{2} - 2])$ .

$2 \frac{{(x^{2} - 2)}^{2} ((x^{2} - 2) (3 x^{2} + 54 x + 6)) + {(x^{2} - 2)}^{2} (- 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{6}}$

Step 3.5.2.3

Factor ${(x^{2} - 2)}^{2}$ out of ${(x^{2} - 2)}^{2} ((x^{2} - 2) (3 x^{2} + 54 x + 6)) + {(x^{2} - 2)}^{2} (- 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2} - 2]))$ .

$2 \frac{{(x^{2} - 2)}^{2} ((x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{6}}$

Step 3.6

Cancel the common factors.

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Step 3.6.1

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

$2 \frac{{(x^{2} - 2)}^{2} ((x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{2} {(x^{2} - 2)}^{4}}$

Step 3.6.2

Cancel the common factor.

$2 \frac{{(x^{2} - 2)}^{2} ((x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{2} {(x^{2} - 2)}^{4}}$

Step 3.6.3

Rewrite the expression.

$2 \frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{4}}$

Step 3.7

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

$2 \frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{4}}$

Step 3.8

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

$2 \frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (2 x + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{4}}$

Step 3.9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 \frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (2 x + 0)}{{(x^{2} - 2)}^{4}}$

Step 3.10

Combine fractions.

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Step 3.10.1

Add $2 x$ and $0$ .

$2 \frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 3 (x^{3} + 27 x^{2} + 6 x + 18) (2 x)}{{(x^{2} - 2)}^{4}}$

Step 3.10.2

Multiply $2$ by $- 3$ .

$2 \frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 6 (x^{3} + 27 x^{2} + 6 x + 18) x}{{(x^{2} - 2)}^{4}}$

Step 3.10.3

Combine $2$ and $\frac{(x^{2} - 2) (3 x^{2} + 54 x + 6) - 6 (x^{3} + 27 x^{2} + 6 x + 18) x}{{(x^{2} - 2)}^{4}}$ .

$\frac{2 ((x^{2} - 2) (3 x^{2} + 54 x + 6) - 6 (x^{3} + 27 x^{2} + 6 x + 18) x)}{{(x^{2} - 2)}^{4}}$

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

$\frac{2 ((x^{2} - 2) (3 x^{2} + 54 x + 6) + (- 6 x^{3} - 6 (27 x^{2}) - 6 (6 x) - 6 \cdot 18) x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.2

Apply the distributive property.

$\frac{2 ((x^{2} - 2) (3 x^{2} + 54 x + 6) - 6 x^{3} x - 6 (27 x^{2}) x - 6 (6 x) x - 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.3

Apply the distributive property.

$\frac{2 ((x^{2} - 2) (3 x^{2} + 54 x + 6)) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4

Simplify the numerator.

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Step 3.11.4.1

Simplify each term.

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Step 3.11.4.1.1

Expand $(x^{2} - 2) (3 x^{2} + 54 x + 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 (x^{2} (3 x^{2}) + x^{2} (54 x) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2

Simplify each term.

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Step 3.11.4.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{2 (3 x^{2} x^{2} + x^{2} (54 x) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.2

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

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Step 3.11.4.1.2.2.1

Move $x^{2}$ .

$\frac{2 (3 (x^{2} x^{2}) + x^{2} (54 x) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.2.2

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

$\frac{2 (3 x^{2 + 2} + x^{2} (54 x) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.2.3

Add $2$ and $2$ .

$\frac{2 (3 x^{4} + x^{2} (54 x) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.3

Rewrite using the commutative property of multiplication.

$\frac{2 (3 x^{4} + 54 x^{2} x + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.4

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

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Step 3.11.4.1.2.4.1

Move $x$ .

$\frac{2 (3 x^{4} + 54 (x \cdot x^{2}) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.4.2

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

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Step 3.11.4.1.2.4.2.1

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

$\frac{2 (3 x^{4} + 54 (x^{1} x^{2}) + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.4.2.2

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

$\frac{2 (3 x^{4} + 54 x^{1 + 2} + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.4.3

Add $1$ and $2$ .

$\frac{2 (3 x^{4} + 54 x^{3} + x^{2} \cdot 6 - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.5

Move $6$ to the left of $x^{2}$ .

$\frac{2 (3 x^{4} + 54 x^{3} + 6 \cdot x^{2} - 2 (3 x^{2}) - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.6

Multiply $3$ by $- 2$ .

$\frac{2 (3 x^{4} + 54 x^{3} + 6 x^{2} - 6 x^{2} - 2 (54 x) - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.7

Multiply $54$ by $- 2$ .

$\frac{2 (3 x^{4} + 54 x^{3} + 6 x^{2} - 6 x^{2} - 108 x - 2 \cdot 6) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.2.8

Multiply $- 2$ by $6$ .

$\frac{2 (3 x^{4} + 54 x^{3} + 6 x^{2} - 6 x^{2} - 108 x - 12) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.3

Combine the opposite terms in $3 x^{4} + 54 x^{3} + 6 x^{2} - 6 x^{2} - 108 x - 12$ .

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Step 3.11.4.1.3.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$\frac{2 (3 x^{4} + 54 x^{3} + 0 - 108 x - 12) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.3.2

Add $3 x^{4} + 54 x^{3}$ and $0$ .

$\frac{2 (3 x^{4} + 54 x^{3} - 108 x - 12) + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.4

Apply the distributive property.

$\frac{2 (3 x^{4}) + 2 (54 x^{3}) + 2 (- 108 x) + 2 \cdot - 12 + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.5

Simplify.

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Step 3.11.4.1.5.1

Multiply $3$ by $2$ .

$\frac{6 x^{4} + 2 (54 x^{3}) + 2 (- 108 x) + 2 \cdot - 12 + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.5.2

Multiply $54$ by $2$ .

$\frac{6 x^{4} + 108 x^{3} + 2 (- 108 x) + 2 \cdot - 12 + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.5.3

Multiply $- 108$ by $2$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x + 2 \cdot - 12 + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.5.4

Multiply $2$ by $- 12$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 + 2 (- 6 x^{3} x) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.6

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

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Step 3.11.4.1.6.1

Move $x$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 + 2 (- 6 (x \cdot x^{3})) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.6.2

Multiply $x$ by $x^{3}$ .

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Step 3.11.4.1.6.2.1

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

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 + 2 (- 6 (x^{1} x^{3})) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.6.2.2

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

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 + 2 (- 6 x^{1 + 3}) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.6.3

Add $1$ and $3$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 + 2 (- 6 x^{4}) + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.7

Multiply $- 6$ by $2$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} + 2 (- 6 (27 x^{2}) x) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.8

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

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Step 3.11.4.1.8.1

Move $x$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} + 2 (- 6 (27 (x \cdot x^{2}))) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.8.2

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

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Step 3.11.4.1.8.2.1

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

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} + 2 (- 6 (27 (x^{1} x^{2}))) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.8.2.2

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

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} + 2 (- 6 (27 x^{1 + 2})) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.8.3

Add $1$ and $2$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} + 2 (- 6 (27 x^{3})) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.9

Multiply $27$ by $- 6$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} + 2 (- 162 x^{3}) + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.10

Multiply $- 162$ by $2$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} + 2 (- 6 (6 x) x) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.11

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

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Step 3.11.4.1.11.1

Move $x$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} + 2 (- 6 (6 (x \cdot x))) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.11.2

Multiply $x$ by $x$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} + 2 (- 6 (6 x^{2})) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.12

Multiply $6$ by $- 6$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} + 2 (- 36 x^{2}) + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.13

Multiply $- 36$ by $2$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} - 72 x^{2} + 2 (- 6 \cdot 18 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.14

Multiply $- 6$ by $18$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} - 72 x^{2} + 2 (- 108 x)}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.1.15

Multiply $- 108$ by $2$ .

$\frac{6 x^{4} + 108 x^{3} - 216 x - 24 - 12 x^{4} - 324 x^{3} - 72 x^{2} - 216 x}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.2

Subtract $12 x^{4}$ from $6 x^{4}$ .

$\frac{- 6 x^{4} + 108 x^{3} - 216 x - 24 - 324 x^{3} - 72 x^{2} - 216 x}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.3

Subtract $324 x^{3}$ from $108 x^{3}$ .

$\frac{- 6 x^{4} - 216 x^{3} - 216 x - 24 - 72 x^{2} - 216 x}{{(x^{2} - 2)}^{4}}$

Step 3.11.4.4

Subtract $216 x$ from $- 216 x$ .

$\frac{- 6 x^{4} - 216 x^{3} - 432 x - 24 - 72 x^{2}}{{(x^{2} - 2)}^{4}}$

Step 3.11.5

Reorder terms.

$\frac{- 6 x^{4} - 216 x^{3} - 72 x^{2} - 432 x - 24}{{(x^{2} - 2)}^{4}}$

Step 3.11.6

Factor $6$ out of $- 6 x^{4} - 216 x^{3} - 72 x^{2} - 432 x - 24$ .

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Step 3.11.6.1

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

$\frac{6 (- x^{4}) - 216 x^{3} - 72 x^{2} - 432 x - 24}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.2

Factor $6$ out of $- 216 x^{3}$ .

$\frac{6 (- x^{4}) + 6 (- 36 x^{3}) - 72 x^{2} - 432 x - 24}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.3

Factor $6$ out of $- 72 x^{2}$ .

$\frac{6 (- x^{4}) + 6 (- 36 x^{3}) + 6 (- 12 x^{2}) - 432 x - 24}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.4

Factor $6$ out of $- 432 x$ .

$\frac{6 (- x^{4}) + 6 (- 36 x^{3}) + 6 (- 12 x^{2}) + 6 (- 72 x) - 24}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.5

Factor $6$ out of $- 24$ .

$\frac{6 (- x^{4}) + 6 (- 36 x^{3}) + 6 (- 12 x^{2}) + 6 (- 72 x) + 6 (- 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.6

Factor $6$ out of $6 (- x^{4}) + 6 (- 36 x^{3})$ .

$\frac{6 (- x^{4} - 36 x^{3}) + 6 (- 12 x^{2}) + 6 (- 72 x) + 6 (- 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.7

Factor $6$ out of $6 (- x^{4} - 36 x^{3}) + 6 (- 12 x^{2})$ .

$\frac{6 (- x^{4} - 36 x^{3} - 12 x^{2}) + 6 (- 72 x) + 6 (- 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.8

Factor $6$ out of $6 (- x^{4} - 36 x^{3} - 12 x^{2}) + 6 (- 72 x)$ .

$\frac{6 (- x^{4} - 36 x^{3} - 12 x^{2} - 72 x) + 6 (- 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.6.9

Factor $6$ out of $6 (- x^{4} - 36 x^{3} - 12 x^{2} - 72 x) + 6 (- 4)$ .

$\frac{6 (- x^{4} - 36 x^{3} - 12 x^{2} - 72 x - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.7

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

$\frac{6 (- (x^{4}) - 36 x^{3} - 12 x^{2} - 72 x - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.8

Factor $- 1$ out of $- 36 x^{3}$ .

$\frac{6 (- (x^{4}) - (36 x^{3}) - 12 x^{2} - 72 x - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.9

Factor $- 1$ out of $- (x^{4}) - (36 x^{3})$ .

$\frac{6 (- (x^{4} + 36 x^{3}) - 12 x^{2} - 72 x - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.10

Factor $- 1$ out of $- 12 x^{2}$ .

$\frac{6 (- (x^{4} + 36 x^{3}) - (12 x^{2}) - 72 x - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.11

Factor $- 1$ out of $- (x^{4} + 36 x^{3}) - (12 x^{2})$ .

$\frac{6 (- (x^{4} + 36 x^{3} + 12 x^{2}) - 72 x - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.12

Factor $- 1$ out of $- 72 x$ .

$\frac{6 (- (x^{4} + 36 x^{3} + 12 x^{2}) - (72 x) - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.13

Factor $- 1$ out of $- (x^{4} + 36 x^{3} + 12 x^{2}) - (72 x)$ .

$\frac{6 (- (x^{4} + 36 x^{3} + 12 x^{2} + 72 x) - 4)}{{(x^{2} - 2)}^{4}}$

Step 3.11.14

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{6 (- (x^{4} + 36 x^{3} + 12 x^{2} + 72 x) - 1 (4))}{{(x^{2} - 2)}^{4}}$

Step 3.11.15

Factor $- 1$ out of $- (x^{4} + 36 x^{3} + 12 x^{2} + 72 x) - 1 (4)$ .

$\frac{6 (- (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4))}{{(x^{2} - 2)}^{4}}$

Step 3.11.16

Rewrite $- (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4)$ as $- 1 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4)$ .

$\frac{6 (- 1 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4))}{{(x^{2} - 2)}^{4}}$

Step 3.11.17

Move the negative in front of the fraction.

$f''' (x) = - \frac{6 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4)}{{(x^{2} - 2)}^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4)}{{(x^{2} - 2)}^{4}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4}{{(x^{2} - 2)}^{4}}]$ .

$- 6 \frac{d}{d x} [\frac{x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4}{{(x^{2} - 2)}^{4}}]$

Step 4.2

$- 6 \frac{{(x^{2} - 2)}^{4} \frac{d}{d x} [x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4] - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{({(x^{2} - 2)}^{4})}^{2}}$

Step 4.3

Differentiate.

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Step 4.3.1

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

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Step 4.3.1.1

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

$- 6 \frac{{(x^{2} - 2)}^{4} \frac{d}{d x} [x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4] - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{4 \cdot 2}}$

Step 4.3.1.2

Multiply $4$ by $2$ .

$- 6 \frac{{(x^{2} - 2)}^{4} \frac{d}{d x} [x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4] - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.2

By the Sum Rule, the derivative of $x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [36 x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [36 x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + \frac{d}{d x} [36 x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.4

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{3}$ with respect to $x$ is $36 \frac{d}{d x} [x^{3}]$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 36 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.5

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

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 36 (3 x^{2}) + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.6

Multiply $3$ by $36$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.7

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.8

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

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 12 (2 x) + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.9

Multiply $2$ by $12$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + \frac{d}{d x} [72 x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.10

Since $72$ is constant with respect to $x$ , the derivative of $72 x$ with respect to $x$ is $72 \frac{d}{d x} [x]$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72 \frac{d}{d x} [x] + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.11

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

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72 \cdot 1 + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.12

Multiply $72$ by $1$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72 + \frac{d}{d x} [4]) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.13

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

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72 + 0) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.3.14

Add $4 x^{3} + 108 x^{2} + 24 x + 72$ and $0$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) \frac{d}{d x} [{(x^{2} - 2)}^{4}]}{{(x^{2} - 2)}^{8}}$

Step 4.4

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

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Step 4.4.1

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

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{8}}$

Step 4.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (4 u^{3} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{8}}$

Step 4.4.3

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

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72) - (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (4 {(x^{2} - 2)}^{3} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{8}}$

Step 4.5

Simplify with factoring out.

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Step 4.5.1

Multiply $4$ by $- 1$ .

$- 6 \frac{{(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) ({(x^{2} - 2)}^{3} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{8}}$

Step 4.5.2

Factor ${(x^{2} - 2)}^{3}$ out of ${(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) ({(x^{2} - 2)}^{3} \frac{d}{d x} [x^{2} - 2])$ .

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Step 4.5.2.1

Factor ${(x^{2} - 2)}^{3}$ out of ${(x^{2} - 2)}^{4} (4 x^{3} + 108 x^{2} + 24 x + 72)$ .

$- 6 \frac{{(x^{2} - 2)}^{3} ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72)) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) ({(x^{2} - 2)}^{3} \frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{8}}$

Step 4.5.2.2

Factor ${(x^{2} - 2)}^{3}$ out of $- 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) ({(x^{2} - 2)}^{3} \frac{d}{d x} [x^{2} - 2])$ .

$- 6 \frac{{(x^{2} - 2)}^{3} ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72)) + {(x^{2} - 2)}^{3} (- 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{8}}$

Step 4.5.2.3

Factor ${(x^{2} - 2)}^{3}$ out of ${(x^{2} - 2)}^{3} ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72)) + {(x^{2} - 2)}^{3} (- 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2} - 2]))$ .

$- 6 \frac{{(x^{2} - 2)}^{3} ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{8}}$

Step 4.6

Cancel the common factors.

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Step 4.6.1

Factor ${(x^{2} - 2)}^{3}$ out of ${(x^{2} - 2)}^{8}$ .

$- 6 \frac{{(x^{2} - 2)}^{3} ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{3} {(x^{2} - 2)}^{5}}$

Step 4.6.2

Cancel the common factor.

$- 6 \frac{{(x^{2} - 2)}^{3} ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2} - 2]))}{{(x^{2} - 2)}^{3} {(x^{2} - 2)}^{5}}$

Step 4.6.3

Rewrite the expression.

$- 6 \frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2} - 2])}{{(x^{2} - 2)}^{5}}$

Step 4.7

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

$- 6 \frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{5}}$

Step 4.8

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

$- 6 \frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (2 x + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{5}}$

Step 4.9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- 6 \frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (2 x + 0)}{{(x^{2} - 2)}^{5}}$

Step 4.10

Combine fractions.

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Step 4.10.1

Add $2 x$ and $0$ .

$- 6 \frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 4 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) (2 x)}{{(x^{2} - 2)}^{5}}$

Step 4.10.2

Multiply $2$ by $- 4$ .

$- 6 \frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 8 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) x}{{(x^{2} - 2)}^{5}}$

Step 4.10.3

Combine $- 6$ and $\frac{(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 8 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) x}{{(x^{2} - 2)}^{5}}$ .

$\frac{- 6 ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 8 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) x)}{{(x^{2} - 2)}^{5}}$

Step 4.10.4

Move the negative in front of the fraction.

$- \frac{6 ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 8 (x^{4} + 36 x^{3} + 12 x^{2} + 72 x + 4) x)}{{(x^{2} - 2)}^{5}}$

Step 4.11

Simplify.

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Step 4.11.1

Apply the distributive property.

$- \frac{6 ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) + (- 8 x^{4} - 8 (36 x^{3}) - 8 (12 x^{2}) - 8 (72 x) - 8 \cdot 4) x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.2

Apply the distributive property.

$- \frac{6 ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72) - 8 x^{4} x - 8 (36 x^{3}) x - 8 (12 x^{2}) x - 8 (72 x) x - 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.3

Apply the distributive property.

$- \frac{6 ((x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72)) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4

Simplify the numerator.

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Step 4.11.4.1

Simplify each term.

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Step 4.11.4.1.1

Expand $(x^{2} - 2) (4 x^{3} + 108 x^{2} + 24 x + 72)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{6 (x^{2} (4 x^{3}) + x^{2} (108 x^{2}) + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2

Simplify each term.

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Step 4.11.4.1.2.1

Rewrite using the commutative property of multiplication.

$- \frac{6 (4 x^{2} x^{3} + x^{2} (108 x^{2}) + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.2

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

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Step 4.11.4.1.2.2.1

Move $x^{3}$ .

$- \frac{6 (4 (x^{3} x^{2}) + x^{2} (108 x^{2}) + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.2.2

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

$- \frac{6 (4 x^{3 + 2} + x^{2} (108 x^{2}) + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.2.3

Add $3$ and $2$ .

$- \frac{6 (4 x^{5} + x^{2} (108 x^{2}) + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.3

Rewrite using the commutative property of multiplication.

$- \frac{6 (4 x^{5} + 108 x^{2} x^{2} + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.4

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

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Step 4.11.4.1.2.4.1

Move $x^{2}$ .

$- \frac{6 (4 x^{5} + 108 (x^{2} x^{2}) + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.4.2

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

$- \frac{6 (4 x^{5} + 108 x^{2 + 2} + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.4.3

Add $2$ and $2$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + x^{2} (24 x) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.5

Rewrite using the commutative property of multiplication.

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{2} x + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.6

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

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Step 4.11.4.1.2.6.1

Move $x$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 (x \cdot x^{2}) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.6.2

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

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Step 4.11.4.1.2.6.2.1

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

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 (x^{1} x^{2}) + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.6.2.2

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

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{1 + 2} + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.6.3

Add $1$ and $2$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{3} + x^{2} \cdot 72 - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.7

Move $72$ to the left of $x^{2}$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{3} + 72 \cdot x^{2} - 2 (4 x^{3}) - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.8

Multiply $4$ by $- 2$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{3} + 72 x^{2} - 8 x^{3} - 2 (108 x^{2}) - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.9

Multiply $108$ by $- 2$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{3} + 72 x^{2} - 8 x^{3} - 216 x^{2} - 2 (24 x) - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.10

Multiply $24$ by $- 2$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{3} + 72 x^{2} - 8 x^{3} - 216 x^{2} - 48 x - 2 \cdot 72) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.2.11

Multiply $- 2$ by $72$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 24 x^{3} + 72 x^{2} - 8 x^{3} - 216 x^{2} - 48 x - 144) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.3

Subtract $8 x^{3}$ from $24 x^{3}$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 16 x^{3} + 72 x^{2} - 216 x^{2} - 48 x - 144) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.4

Subtract $216 x^{2}$ from $72 x^{2}$ .

$- \frac{6 (4 x^{5} + 108 x^{4} + 16 x^{3} - 144 x^{2} - 48 x - 144) + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.5

Apply the distributive property.

$- \frac{6 (4 x^{5}) + 6 (108 x^{4}) + 6 (16 x^{3}) + 6 (- 144 x^{2}) + 6 (- 48 x) + 6 \cdot - 144 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.6

Simplify.

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Step 4.11.4.1.6.1

Multiply $4$ by $6$ .

$- \frac{24 x^{5} + 6 (108 x^{4}) + 6 (16 x^{3}) + 6 (- 144 x^{2}) + 6 (- 48 x) + 6 \cdot - 144 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.6.2

Multiply $108$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 6 (16 x^{3}) + 6 (- 144 x^{2}) + 6 (- 48 x) + 6 \cdot - 144 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.6.3

Multiply $16$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} + 6 (- 144 x^{2}) + 6 (- 48 x) + 6 \cdot - 144 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.6.4

Multiply $- 144$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} + 6 (- 48 x) + 6 \cdot - 144 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.6.5

Multiply $- 48$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x + 6 \cdot - 144 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.6.6

Multiply $6$ by $- 144$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 + 6 (- 8 x^{4} x) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.7

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

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Step 4.11.4.1.7.1

Move $x$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 + 6 (- 8 (x \cdot x^{4})) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.7.2

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

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Step 4.11.4.1.7.2.1

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

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 + 6 (- 8 (x^{1} x^{4})) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.7.2.2

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

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 + 6 (- 8 x^{1 + 4}) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.7.3

Add $1$ and $4$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 + 6 (- 8 x^{5}) + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.8

Multiply $- 8$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} + 6 (- 8 (36 x^{3}) x) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.9

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

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Step 4.11.4.1.9.1

Move $x$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} + 6 (- 8 (36 (x \cdot x^{3}))) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.9.2

Multiply $x$ by $x^{3}$ .

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Step 4.11.4.1.9.2.1

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

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} + 6 (- 8 (36 (x^{1} x^{3}))) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.9.2.2

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

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} + 6 (- 8 (36 x^{1 + 3})) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.9.3

Add $1$ and $3$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} + 6 (- 8 (36 x^{4})) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.10

Multiply $36$ by $- 8$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} + 6 (- 288 x^{4}) + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.11

Multiply $- 288$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} + 6 (- 8 (12 x^{2}) x) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.12

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

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Step 4.11.4.1.12.1

Move $x$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} + 6 (- 8 (12 (x \cdot x^{2}))) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.12.2

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

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Step 4.11.4.1.12.2.1

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

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} + 6 (- 8 (12 (x^{1} x^{2}))) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.12.2.2

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

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} + 6 (- 8 (12 x^{1 + 2})) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.12.3

Add $1$ and $2$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} + 6 (- 8 (12 x^{3})) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.13

Multiply $12$ by $- 8$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} + 6 (- 96 x^{3}) + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.14

Multiply $- 96$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} + 6 (- 8 (72 x) x) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.15

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

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Step 4.11.4.1.15.1

Move $x$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} + 6 (- 8 (72 (x \cdot x))) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.15.2

Multiply $x$ by $x$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} + 6 (- 8 (72 x^{2})) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.16

Multiply $72$ by $- 8$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} + 6 (- 576 x^{2}) + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.17

Multiply $- 576$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} - 3456 x^{2} + 6 (- 8 \cdot 4 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.18

Multiply $- 8$ by $4$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} - 3456 x^{2} + 6 (- 32 x)}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.1.19

Multiply $- 32$ by $6$ .

$- \frac{24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 48 x^{5} - 1728 x^{4} - 576 x^{3} - 3456 x^{2} - 192 x}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.2

Subtract $48 x^{5}$ from $24 x^{5}$ .

$- \frac{- 24 x^{5} + 648 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 1728 x^{4} - 576 x^{3} - 3456 x^{2} - 192 x}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.3

Subtract $1728 x^{4}$ from $648 x^{4}$ .

$- \frac{- 24 x^{5} - 1080 x^{4} + 96 x^{3} - 864 x^{2} - 288 x - 864 - 576 x^{3} - 3456 x^{2} - 192 x}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.4

Subtract $576 x^{3}$ from $96 x^{3}$ .

$- \frac{- 24 x^{5} - 1080 x^{4} - 480 x^{3} - 864 x^{2} - 288 x - 864 - 3456 x^{2} - 192 x}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.5

Subtract $3456 x^{2}$ from $- 864 x^{2}$ .

$- \frac{- 24 x^{5} - 1080 x^{4} - 480 x^{3} - 4320 x^{2} - 288 x - 864 - 192 x}{{(x^{2} - 2)}^{5}}$

Step 4.11.4.6

Subtract $192 x$ from $- 288 x$ .

$- \frac{- 24 x^{5} - 1080 x^{4} - 480 x^{3} - 4320 x^{2} - 480 x - 864}{{(x^{2} - 2)}^{5}}$

Step 4.11.5

Factor $24$ out of $- 24 x^{5} - 1080 x^{4} - 480 x^{3} - 4320 x^{2} - 480 x - 864$ .

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Step 4.11.5.1

Factor $24$ out of $- 24 x^{5}$ .

$- \frac{24 (- x^{5}) - 1080 x^{4} - 480 x^{3} - 4320 x^{2} - 480 x - 864}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.2

Factor $24$ out of $- 1080 x^{4}$ .

$- \frac{24 (- x^{5}) + 24 (- 45 x^{4}) - 480 x^{3} - 4320 x^{2} - 480 x - 864}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.3

Factor $24$ out of $- 480 x^{3}$ .

$- \frac{24 (- x^{5}) + 24 (- 45 x^{4}) + 24 (- 20 x^{3}) - 4320 x^{2} - 480 x - 864}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.4

Factor $24$ out of $- 4320 x^{2}$ .

$- \frac{24 (- x^{5}) + 24 (- 45 x^{4}) + 24 (- 20 x^{3}) + 24 (- 180 x^{2}) - 480 x - 864}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.5

Factor $24$ out of $- 480 x$ .

$- \frac{24 (- x^{5}) + 24 (- 45 x^{4}) + 24 (- 20 x^{3}) + 24 (- 180 x^{2}) + 24 (- 20 x) - 864}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.6

Factor $24$ out of $- 864$ .

$- \frac{24 (- x^{5}) + 24 (- 45 x^{4}) + 24 (- 20 x^{3}) + 24 (- 180 x^{2}) + 24 (- 20 x) + 24 (- 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.7

Factor $24$ out of $24 (- x^{5}) + 24 (- 45 x^{4})$ .

$- \frac{24 (- x^{5} - 45 x^{4}) + 24 (- 20 x^{3}) + 24 (- 180 x^{2}) + 24 (- 20 x) + 24 (- 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.8

Factor $24$ out of $24 (- x^{5} - 45 x^{4}) + 24 (- 20 x^{3})$ .

$- \frac{24 (- x^{5} - 45 x^{4} - 20 x^{3}) + 24 (- 180 x^{2}) + 24 (- 20 x) + 24 (- 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.9

Factor $24$ out of $24 (- x^{5} - 45 x^{4} - 20 x^{3}) + 24 (- 180 x^{2})$ .

$- \frac{24 (- x^{5} - 45 x^{4} - 20 x^{3} - 180 x^{2}) + 24 (- 20 x) + 24 (- 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.10

Factor $24$ out of $24 (- x^{5} - 45 x^{4} - 20 x^{3} - 180 x^{2}) + 24 (- 20 x)$ .

$- \frac{24 (- x^{5} - 45 x^{4} - 20 x^{3} - 180 x^{2} - 20 x) + 24 (- 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.5.11

Factor $24$ out of $24 (- x^{5} - 45 x^{4} - 20 x^{3} - 180 x^{2} - 20 x) + 24 (- 36)$ .

$- \frac{24 (- x^{5} - 45 x^{4} - 20 x^{3} - 180 x^{2} - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.6

Factor $- 1$ out of $- x^{5}$ .

$- \frac{24 (- (x^{5}) - 45 x^{4} - 20 x^{3} - 180 x^{2} - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.7

Factor $- 1$ out of $- 45 x^{4}$ .

$- \frac{24 (- (x^{5}) - (45 x^{4}) - 20 x^{3} - 180 x^{2} - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.8

Factor $- 1$ out of $- (x^{5}) - (45 x^{4})$ .

$- \frac{24 (- (x^{5} + 45 x^{4}) - 20 x^{3} - 180 x^{2} - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.9

Factor $- 1$ out of $- 20 x^{3}$ .

$- \frac{24 (- (x^{5} + 45 x^{4}) - (20 x^{3}) - 180 x^{2} - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.10

Factor $- 1$ out of $- (x^{5} + 45 x^{4}) - (20 x^{3})$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3}) - 180 x^{2} - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.11

Factor $- 1$ out of $- 180 x^{2}$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3}) - (180 x^{2}) - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.12

Factor $- 1$ out of $- (x^{5} + 45 x^{4} + 20 x^{3}) - (180 x^{2})$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2}) - 20 x - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.13

Factor $- 1$ out of $- 20 x$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2}) - (20 x) - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.14

Factor $- 1$ out of $- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2}) - (20 x)$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x) - 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.15

Rewrite $- 36$ as $- 1 (36)$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x) - 1 (36))}{{(x^{2} - 2)}^{5}}$

Step 4.11.16

Factor $- 1$ out of $- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x) - 1 (36)$ .

$- \frac{24 (- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36))}{{(x^{2} - 2)}^{5}}$

Step 4.11.17

Rewrite $- (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)$ as $- 1 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)$ .

$- \frac{24 (- 1 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36))}{{(x^{2} - 2)}^{5}}$

Step 4.11.18

Move the negative in front of the fraction.

$- - \frac{24 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.19

Multiply $- 1$ by $- 1$ .

$1 \frac{24 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)}{{(x^{2} - 2)}^{5}}$

Step 4.11.20

Multiply $\frac{24 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)}{{(x^{2} - 2)}^{5}}$ by $1$ .

$f^{4} (x) = \frac{24 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)}{{(x^{2} - 2)}^{5}}$

Step 5

The fourth derivative of $g (x)$ with respect to $x$ is $\frac{24 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)}{{(x^{2} - 2)}^{5}}$ .

$\frac{24 (x^{5} + 45 x^{4} + 20 x^{3} + 180 x^{2} + 20 x + 36)}{{(x^{2} - 2)}^{5}}$