Calculus Examples

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

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 $f (x) = x^{2} - 1$ and $g (x) = x^{2} + 1$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.2.4.2

Move $2$ to the left of $x^{2} + 1$ .

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

Step 1.2.5

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $2 x$ and $0$ .

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

Step 1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Apply the distributive property.

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

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Combine the opposite terms in $2 x^{2} x + 2 \cdot 1 x - 2 x^{2} x - 2 \cdot - 1 x$ .

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Step 1.3.5.1.1

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

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

Step 1.3.5.1.2

Add $2 \cdot 1 x$ and $0$ .

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

Step 1.3.5.2

Simplify each term.

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Step 1.3.5.2.1

Multiply $2$ by $1$ .

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

Step 1.3.5.2.2

Multiply $- 2$ by $- 1$ .

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

Step 1.3.5.3

Add $2 x$ and $2 x$ .

$f' (x) = \frac{4 x}{{(x^{2} + 1)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x}{{(x^{2} + 1)}^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x}{{(x^{2} + 1)}^{2}}]$ .

$f'' (x) = 4 \frac{d}{d x} (\frac{x}{{(x^{2} + 1)}^{2}})$

Step 2.2

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{({(x^{2} + 1)}^{2})}^{2}})$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Multiply the exponents in ${({(x^{2} + 1)}^{2})}^{2}$ .

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Step 2.3.1.1

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

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{2 \cdot 2}})$

Step 2.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{4}})$

Step 2.3.2

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

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{4}})$

Step 2.3.3

Multiply ${(x^{2} + 1)}^{2}$ by $1$ .

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{4}})$

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 $g (x) = x^{2} + 1$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{4}})$

Step 2.4.2

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

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} - x (2 u \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{4}})$

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} - x (2 (x^{2} + 1) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{4}})$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = 4 (\frac{{(x^{2} + 1)}^{2} - 2 x ((x^{2} + 1) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{4}})$

Step 2.5.2

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{2} - 2 x ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

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Step 2.5.2.1

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

$f'' (x) = 4 (\frac{(x^{2} + 1) (x^{2} + 1) - 2 x ((x^{2} + 1) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{4}})$

Step 2.5.2.2

Factor $x^{2} + 1$ out of $- 2 x ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

$f'' (x) = 4 (\frac{(x^{2} + 1) (x^{2} + 1) + (x^{2} + 1) (- 2 x (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{4}})$

Step 2.5.2.3

Factor $x^{2} + 1$ out of $(x^{2} + 1) (x^{2} + 1) + (x^{2} + 1) (- 2 x (\frac{d}{d x} [x^{2} + 1]))$ .

$f'' (x) = 4 (\frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{4}})$

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$f'' (x) = 4 (\frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1)))}{(x^{2} + 1) {(x^{2} + 1)}^{3}})$

Step 2.6.2

Cancel the common factor.

$f'' (x) = 4 (\frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1)))}{(x^{2} + 1) {(x^{2} + 1)}^{3}})$

Step 2.6.3

Rewrite the expression.

$f'' (x) = 4 (\frac{x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{3}})$

Step 2.7

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$f'' (x) = 4 (\frac{x^{2} + 1 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{3}})$

Step 2.8

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

$f'' (x) = 4 (\frac{x^{2} + 1 - 2 x (2 x + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{3}})$

Step 2.9

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

$f'' (x) = 4 (\frac{x^{2} + 1 - 2 x (2 x + 0)}{{(x^{2} + 1)}^{3}})$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

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

Step 2.10.2

Multiply $2$ by $- 2$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Add $1$ and $1$ .

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

Step 2.15

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

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

Step 2.16

Combine $4$ and $\frac{- 3 x^{2} + 1}{{(x^{2} + 1)}^{3}}$ .

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

Step 2.17

Simplify.

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Step 2.17.1

Apply the distributive property.

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

Step 2.17.2

Simplify each term.

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Step 2.17.2.1

Multiply $- 3$ by $4$ .

$f'' (x) = \frac{- 12 x^{2} + 4 \cdot 1}{{(x^{2} + 1)}^{3}}$

Step 2.17.2.2

Multiply $4$ by $1$ .

$f'' (x) = \frac{- 12 x^{2} + 4}{{(x^{2} + 1)}^{3}}$

Step 2.17.3

Factor $4$ out of $- 12 x^{2} + 4$ .

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Step 2.17.3.1

Factor $4$ out of $- 12 x^{2}$ .

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

Step 2.17.3.2

Factor $4$ out of $4$ .

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

Step 2.17.3.3

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

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

Step 2.17.4

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

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

Step 2.17.5

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

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

Step 2.17.6

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

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

Step 2.17.7

Rewrite $- (3 x^{2} - 1)$ as $- 1 (3 x^{2} - 1)$ .

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

Step 2.17.8

Move the negative in front of the fraction.

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

Step 3

Find the third derivative.

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Step 3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{3 x^{2} - 1}{{(x^{2} + 1)}^{3}}]$ .

$f''' (x) = - 4 \frac{d}{d x} \frac{3 x^{2} - 1}{{(x^{2} + 1)}^{3}}$

Step 3.2

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} \frac{d}{d x} (3 x^{2} - 1) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{({(x^{2} + 1)}^{3})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(x^{2} + 1)}^{3})}^{2}$ .

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Step 3.3.1.1

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

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} \frac{d}{d x} (3 x^{2} - 1) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{3 \cdot 2}}$

Step 3.3.1.2

Multiply $3$ by $2$ .

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} \frac{d}{d x} (3 x^{2} - 1) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.2

By the Sum Rule, the derivative of $3 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 1]$ .

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} (\frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 1)) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.3

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

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} (3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1)) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.4

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

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} (3 (2 x) + \frac{d}{d x} (- 1)) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.5

Multiply $2$ by $3$ .

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} (6 x + \frac{d}{d x} (- 1)) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.6

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

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} (6 x + 0) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $6 x$ and $0$ .

$f''' (x) = - 4 \frac{{(x^{2} + 1)}^{3} (6 x) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

Step 3.3.7.2

Move $6$ to the left of ${(x^{2} + 1)}^{3}$ .

$f''' (x) = - 4 \frac{6 \cdot ({(x^{2} + 1)}^{3} x) - (3 x^{2} - 1) \frac{d}{d x} {(x^{2} + 1)}^{3}}{{(x^{2} + 1)}^{6}}$

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 $g (x) = x^{2} + 1$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - (3 x^{2} - 1) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{6}}$

Step 3.4.2

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

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - (3 x^{2} - 1) (3 u^{2} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{6}}$

Step 3.4.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - (3 x^{2} - 1) (3 {(x^{2} + 1)}^{2} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{6}}$

Step 3.5

Differentiate.

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Step 3.5.1

Multiply $3$ by $- 1$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 3 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{6}}$

Step 3.5.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 3 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1)))}{{(x^{2} + 1)}^{6}}$

Step 3.5.3

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

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 3 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} (2 x + \frac{d}{d x} (1)))}{{(x^{2} + 1)}^{6}}$

Step 3.5.4

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

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 3 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} (2 x + 0))}{{(x^{2} + 1)}^{6}}$

Step 3.5.5

Combine fractions.

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Step 3.5.5.1

Add $2 x$ and $0$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 3 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} (2 x))}{{(x^{2} + 1)}^{6}}$

Step 3.5.5.2

Simplify the expression.

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Step 3.5.5.2.1

Move $2$ to the left of ${(x^{2} + 1)}^{2}$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 3 (3 x^{2} - 1) (2 \cdot ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.5.5.2.2

Multiply $2$ by $- 3$ .

$f''' (x) = - 4 \frac{6 {(x^{2} + 1)}^{3} x - 6 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} x)}{{(x^{2} + 1)}^{6}}$

Step 3.5.5.3

Combine $- 4$ and $\frac{6 {(x^{2} + 1)}^{3} x - 6 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} x)}{{(x^{2} + 1)}^{6}}$ .

$f''' (x) = \frac{- 4 (6 {(x^{2} + 1)}^{3} x - 6 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.5.5.4

Move the negative in front of the fraction.

$f''' (x) = - \frac{(4) (6 {(x^{2} + 1)}^{3} x - 6 (3 x^{2} - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$f''' (x) = - \frac{4 (6 {(x^{2} + 1)}^{3} x + (- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.2

Apply the distributive property.

$f''' (x) = - \frac{4 (6 {(x^{2} + 1)}^{3} x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3

Simplify the numerator.

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Step 3.6.3.1

Use the Binomial Theorem.

$f''' (x) = - \frac{4 (6 ({(x^{2})}^{3} + 3 {(x^{2})}^{2} \cdot 1 + 3 x^{2} \cdot 1^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2

Simplify each term.

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Step 3.6.3.2.1

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

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Step 3.6.3.2.1.1

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

$f''' (x) = - \frac{4 (6 (x^{2 \cdot 3} + 3 {(x^{2})}^{2} \cdot 1 + 3 x^{2} \cdot 1^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.1.2

Multiply $2$ by $3$ .

$f''' (x) = - \frac{4 (6 (x^{6} + 3 {(x^{2})}^{2} \cdot 1 + 3 x^{2} \cdot 1^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.2

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

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Step 3.6.3.2.2.1

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

$f''' (x) = - \frac{4 (6 (x^{6} + 3 x^{2 \cdot 2} \cdot 1 + 3 x^{2} \cdot 1^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.2.2

Multiply $2$ by $2$ .

$f''' (x) = - \frac{4 (6 (x^{6} + 3 x^{4} \cdot 1 + 3 x^{2} \cdot 1^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.3

Multiply $3$ by $1$ .

$f''' (x) = - \frac{4 (6 (x^{6} + 3 x^{4} + 3 x^{2} \cdot 1^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.4

One to any power is one.

$f''' (x) = - \frac{4 (6 (x^{6} + 3 x^{4} + 3 x^{2} \cdot 1 + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.5

Multiply $3$ by $1$ .

$f''' (x) = - \frac{4 (6 (x^{6} + 3 x^{4} + 3 x^{2} + 1^{3}) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.2.6

One to any power is one.

$f''' (x) = - \frac{4 (6 (x^{6} + 3 x^{4} + 3 x^{2} + 1) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.3

Apply the distributive property.

$f''' (x) = - \frac{4 ((6 x^{6} + 6 (3 x^{4}) + 6 (3 x^{2}) + 6 \cdot 1) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.4

Simplify.

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Step 3.6.3.4.1

Multiply $3$ by $6$ .

$f''' (x) = - \frac{4 ((6 x^{6} + 18 x^{4} + 6 (3 x^{2}) + 6 \cdot 1) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.4.2

Multiply $3$ by $6$ .

$f''' (x) = - \frac{4 ((6 x^{6} + 18 x^{4} + 18 x^{2} + 6 \cdot 1) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.4.3

Multiply $6$ by $1$ .

$f''' (x) = - \frac{4 ((6 x^{6} + 18 x^{4} + 18 x^{2} + 6) x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.5

Apply the distributive property.

$f''' (x) = - \frac{4 (6 x^{6} x + 18 x^{4} x + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6

Simplify.

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Step 3.6.3.6.1

Multiply $x^{6}$ by $x$ by adding the exponents.

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Step 3.6.3.6.1.1

Move $x$ .

$f''' (x) = - \frac{4 (6 (x \cdot x^{6}) + 18 x^{4} x + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.1.2

Multiply $x$ by $x^{6}$ .

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Step 3.6.3.6.1.2.1

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

$f''' (x) = - \frac{4 (6 (x \cdot x^{6}) + 18 x^{4} x + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.1.2.2

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

$f''' (x) = - \frac{4 (6 x^{1 + 6} + 18 x^{4} x + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.1.3

Add $1$ and $6$ .

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{4} x + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.2

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

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Step 3.6.3.6.2.1

Move $x$ .

$f''' (x) = - \frac{4 (6 x^{7} + 18 (x \cdot x^{4}) + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.2.2

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

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Step 3.6.3.6.2.2.1

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

$f''' (x) = - \frac{4 (6 x^{7} + 18 (x \cdot x^{4}) + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.2.2.2

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

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{1 + 4} + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.2.3

Add $1$ and $4$ .

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{5} + 18 x^{2} x + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.3

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

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Step 3.6.3.6.3.1

Move $x$ .

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{5} + 18 (x \cdot x^{2}) + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.3.2

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

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Step 3.6.3.6.3.2.1

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

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{5} + 18 (x \cdot x^{2}) + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.3.2.2

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

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{5} + 18 x^{1 + 2} + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.6.3.3

Add $1$ and $2$ .

$f''' (x) = - \frac{4 (6 x^{7} + 18 x^{5} + 18 x^{3} + 6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.7

Apply the distributive property.

$f''' (x) = - \frac{4 (6 x^{7}) + 4 (18 x^{5}) + 4 (18 x^{3}) + 4 (6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.8

Simplify.

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Step 3.6.3.8.1

Multiply $6$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 4 (18 x^{5}) + 4 (18 x^{3}) + 4 (6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.8.2

Multiply $18$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 4 (18 x^{3}) + 4 (6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.8.3

Multiply $18$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 4 (6 x) + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.8.4

Multiply $6$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 6 (3 x^{2}) - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.9

Simplify each term.

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Step 3.6.3.9.1

Multiply $3$ by $- 6$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} - 6 \cdot - 1) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.9.2

Multiply $- 6$ by $- 1$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ({(x^{2} + 1)}^{2} x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.10

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{2} + 1) (x^{2} + 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.11

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 3.6.3.11.1

Apply the distributive property.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{2} (x^{2} + 1) + 1 (x^{2} + 1)) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.11.2

Apply the distributive property.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{2} x^{2} + x^{2} \cdot 1 + 1 (x^{2} + 1)) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.11.3

Apply the distributive property.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.12

Simplify and combine like terms.

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Step 3.6.3.12.1

Simplify each term.

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Step 3.6.3.12.1.1

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

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Step 3.6.3.12.1.1.1

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.12.1.1.2

Add $2$ and $2$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.12.1.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.12.1.3

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{4} + x^{2} + x^{2} + 1 \cdot 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.12.1.4

Multiply $1$ by $1$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{4} + x^{2} + x^{2} + 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.12.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) ((x^{4} + 2 x^{2} + 1) x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.13

Apply the distributive property.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{4} x + 2 x^{2} x + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14

Simplify.

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Step 3.6.3.14.1

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

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Step 3.6.3.14.1.1

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

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Step 3.6.3.14.1.1.1

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{4} x + 2 x^{2} x + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.1.1.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{4 + 1} + 2 x^{2} x + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.1.2

Add $4$ and $1$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{5} + 2 x^{2} x + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.2

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

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Step 3.6.3.14.2.1

Move $x$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{5} + 2 (x \cdot x^{2}) + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.2.2

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

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Step 3.6.3.14.2.2.1

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{5} + 2 (x \cdot x^{2}) + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.2.2.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{5} + 2 x^{1 + 2} + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.2.3

Add $1$ and $2$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{5} + 2 x^{3} + 1 x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.14.3

Multiply $x$ by $1$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 ((- 18 x^{2} + 6) (x^{5} + 2 x^{3} + x))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.15

Expand $(- 18 x^{2} + 6) (x^{5} + 2 x^{3} + x)$ by multiplying each term in the first expression by each term in the second expression.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{2} x^{5} - 18 x^{2} (2 x^{3}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16

Simplify each term.

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Step 3.6.3.16.1

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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Step 3.6.3.16.1.1

Move $x^{5}$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 (x^{5} x^{2}) - 18 x^{2} (2 x^{3}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.1.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{5 + 2} - 18 x^{2} (2 x^{3}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.1.3

Add $5$ and $2$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 18 x^{2} (2 x^{3}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.2

Rewrite using the commutative property of multiplication.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 18 \cdot (2 x^{2} x^{3}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.3

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

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Step 3.6.3.16.3.1

Move $x^{3}$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 18 \cdot (2 (x^{3} x^{2})) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.3.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 18 \cdot (2 x^{3 + 2}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.3.3

Add $3$ and $2$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 18 \cdot (2 x^{5}) - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.4

Multiply $- 18$ by $2$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 36 x^{5} - 18 x^{2} x + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.5

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

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Step 3.6.3.16.5.1

Move $x$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 36 x^{5} - 18 (x \cdot x^{2}) + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.5.2

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

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Step 3.6.3.16.5.2.1

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 36 x^{5} - 18 (x \cdot x^{2}) + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.5.2.2

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

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 36 x^{5} - 18 x^{1 + 2} + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.5.3

Add $1$ and $2$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 36 x^{5} - 18 x^{3} + 6 x^{5} + 6 (2 x^{3}) + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.16.6

Multiply $2$ by $6$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 36 x^{5} - 18 x^{3} + 6 x^{5} + 12 x^{3} + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.17

Add $- 36 x^{5}$ and $6 x^{5}$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 30 x^{5} - 18 x^{3} + 12 x^{3} + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.18

Add $- 18 x^{3}$ and $12 x^{3}$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7} - 30 x^{5} - 6 x^{3} + 6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.19

Apply the distributive property.

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x + 4 (- 18 x^{7}) + 4 (- 30 x^{5}) + 4 (- 6 x^{3}) + 4 (6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.20

Simplify.

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Step 3.6.3.20.1

Multiply $- 18$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x - 72 x^{7} + 4 (- 30 x^{5}) + 4 (- 6 x^{3}) + 4 (6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.20.2

Multiply $- 30$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x - 72 x^{7} - 120 x^{5} + 4 (- 6 x^{3}) + 4 (6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.20.3

Multiply $- 6$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x - 72 x^{7} - 120 x^{5} - 24 x^{3} + 4 (6 x)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.20.4

Multiply $6$ by $4$ .

$f''' (x) = - \frac{24 x^{7} + 72 x^{5} + 72 x^{3} + 24 x - 72 x^{7} - 120 x^{5} - 24 x^{3} + 24 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.21

Subtract $72 x^{7}$ from $24 x^{7}$ .

$f''' (x) = - \frac{- 48 x^{7} + 72 x^{5} + 72 x^{3} + 24 x - 120 x^{5} - 24 x^{3} + 24 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.22

Subtract $120 x^{5}$ from $72 x^{5}$ .

$f''' (x) = - \frac{- 48 x^{7} - 48 x^{5} + 72 x^{3} + 24 x - 24 x^{3} + 24 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.23

Subtract $24 x^{3}$ from $72 x^{3}$ .

$f''' (x) = - \frac{- 48 x^{7} - 48 x^{5} + 48 x^{3} + 24 x + 24 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.24

Add $24 x$ and $24 x$ .

$f''' (x) = - \frac{- 48 x^{7} - 48 x^{5} + 48 x^{3} + 48 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25

Rewrite $- 48 x^{7} - 48 x^{5} + 48 x^{3} + 48 x$ in a factored form.

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Step 3.6.3.25.1

Factor $48 x$ out of $- 48 x^{7} - 48 x^{5} + 48 x^{3} + 48 x$ .

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Step 3.6.3.25.1.1

Factor $48 x$ out of $- 48 x^{7}$ .

$f''' (x) = - \frac{48 x (- x^{6}) - 48 x^{5} + 48 x^{3} + 48 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.1.2

Factor $48 x$ out of $- 48 x^{5}$ .

$f''' (x) = - \frac{48 x (- x^{6}) + 48 x (- x^{4}) + 48 x^{3} + 48 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.1.3

Factor $48 x$ out of $48 x^{3}$ .

$f''' (x) = - \frac{48 x (- x^{6}) + 48 x (- x^{4}) + 48 x (x^{2}) + 48 x}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.1.4

Factor $48 x$ out of $48 x$ .

$f''' (x) = - \frac{48 x (- x^{6}) + 48 x (- x^{4}) + 48 x (x^{2}) + 48 x (1)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.1.5

Factor $48 x$ out of $48 x (- x^{6}) + 48 x (- x^{4})$ .

$f''' (x) = - \frac{48 x (- x^{6} - x^{4}) + 48 x (x^{2}) + 48 x (1)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.1.6

Factor $48 x$ out of $48 x (- x^{6} - x^{4}) + 48 x (x^{2})$ .

$f''' (x) = - \frac{48 x (- x^{6} - x^{4} + x^{2}) + 48 x (1)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.1.7

Factor $48 x$ out of $48 x (- x^{6} - x^{4} + x^{2}) + 48 x (1)$ .

$f''' (x) = - \frac{48 x (- x^{6} - x^{4} + x^{2} + 1)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.2

Factor out the greatest common factor from each group.

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Step 3.6.3.25.2.1

Group the first two terms and the last two terms.

$f''' (x) = - \frac{48 x ((- x^{6} - x^{4}) + x^{2} + 1)}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.2.2

Factor out the greatest common factor (GCF) from each group.

$f''' (x) = - \frac{48 x (x^{4} (- x^{2} - 1) - 1 (- x^{2} - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.3

Factor the polynomial by factoring out the greatest common factor, $- x^{2} - 1$ .

$f''' (x) = - \frac{48 x ((- x^{2} - 1) (x^{4} - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.4

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

$f''' (x) = - \frac{48 x ((- x^{2} - 1) ({(x^{2})}^{2} - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.5

Rewrite $1$ as $1^{2}$ .

$f''' (x) = - \frac{48 x ((- x^{2} - 1) ({(x^{2})}^{2} - 1^{2}))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

$f''' (x) = - \frac{48 x ((- x^{2} - 1) ((x^{2} + 1) (x^{2} - 1)))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.7

Simplify.

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Step 3.6.3.25.7.1

Rewrite $1$ as $1^{2}$ .

$f''' (x) = - \frac{48 x ((- x^{2} - 1) ((x^{2} + 1) (x^{2} - 1^{2})))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.7.2

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 $b = 1$ .

$f''' (x) = - \frac{48 x ((- x^{2} - 1) ((x^{2} + 1) (x + 1) (x - 1)))}{{(x^{2} + 1)}^{6}}$

$f''' (x) = - \frac{48 x ((- x^{2} - 1) (x^{2} + 1) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8

Combine exponents.

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Step 3.6.3.25.8.1

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

$f''' (x) = - \frac{48 x ((- (x^{2}) - 1) (x^{2} + 1) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.2

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

$f''' (x) = - \frac{48 x ((- (x^{2}) - 1 \cdot 1) (x^{2} + 1) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.3

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

$f''' (x) = - \frac{48 x (- (x^{2} + 1) (x^{2} + 1) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.4

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

$f''' (x) = - \frac{48 x (- 1 (x^{2} + 1) (x^{2} + 1) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.5

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

$f''' (x) = - \frac{48 x (- 1 ((x^{2} + 1) (x^{2} + 1)) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.6

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

$f''' (x) = - \frac{48 x (- 1 ((x^{2} + 1) (x^{2} + 1)) (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.7

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

$f''' (x) = - \frac{48 x (- 1 {(x^{2} + 1)}^{1 + 1} (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.8.8

Add $1$ and $1$ .

$f''' (x) = - \frac{48 x (- 1 {(x^{2} + 1)}^{2} (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

$f''' (x) = - \frac{48 x \cdot (- 1 {(x^{2} + 1)}^{2} (x + 1) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.3.25.9

Multiply $- 1$ by $48$ .

$f''' (x) = - \frac{- 48 x {(x^{2} + 1)}^{2} (x + 1) (x - 1)}{{(x^{2} + 1)}^{6}}$

Step 3.6.4

Combine terms.

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Step 3.6.4.1

Cancel the common factor of ${(x^{2} + 1)}^{2}$ and ${(x^{2} + 1)}^{6}$ .

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Step 3.6.4.1.1

Factor ${(x^{2} + 1)}^{2}$ out of $- 48 x {(x^{2} + 1)}^{2} (x + 1) (x - 1)$ .

$f''' (x) = - \frac{{(x^{2} + 1)}^{2} ((- 48 x (x + 1)) (x - 1))}{{(x^{2} + 1)}^{6}}$

Step 3.6.4.1.2

Cancel the common factors.

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Step 3.6.4.1.2.1

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

$f''' (x) = - \frac{{(x^{2} + 1)}^{2} ((- 48 x (x + 1)) (x - 1))}{{(x^{2} + 1)}^{2} {(x^{2} + 1)}^{4}}$

Step 3.6.4.1.2.2

Cancel the common factor.

$f''' (x) = - \frac{{(x^{2} + 1)}^{2} ((- 48 x (x + 1)) (x - 1))}{{(x^{2} + 1)}^{2} {(x^{2} + 1)}^{4}}$

Step 3.6.4.1.2.3

Rewrite the expression.

$f''' (x) = - \frac{(- 48 x (x + 1)) (x - 1)}{{(x^{2} + 1)}^{4}}$

Step 3.6.4.2

Move the negative in front of the fraction.

$f''' (x) = \frac{((48) x (x + 1)) (x - 1)}{{(x^{2} + 1)}^{4}}$

Step 3.6.4.3

Multiply $- 1$ by $- 1$ .

$f''' (x) = 1 (\frac{48 x (x + 1) (x - 1)}{{(x^{2} + 1)}^{4}})$

Step 3.6.4.4

Multiply $\frac{48 x (x + 1) (x - 1)}{{(x^{2} + 1)}^{4}}$ by $1$ .

$f''' (x) = \frac{48 x (x + 1) (x - 1)}{{(x^{2} + 1)}^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $48$ is constant with respect to $x$ , the derivative of $\frac{48 x (x + 1) (x - 1)}{{(x^{2} + 1)}^{4}}$ with respect to $x$ is $48 \frac{d}{d x} [\frac{(x (x + 1)) (x - 1)}{{(x^{2} + 1)}^{4}}]$ .

$f^{4} (x) = 48 \frac{d}{d x} (\frac{(x (x + 1)) (x - 1)}{{(x^{2} + 1)}^{4}})$

Step 4.2

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} \frac{d}{d x} (x (x + 1) (x - 1)) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{({(x^{2} + 1)}^{4})}^{2}})$

Step 4.3

Multiply the exponents in ${({(x^{2} + 1)}^{4})}^{2}$ .

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Step 4.3.1

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} \frac{d}{d x} (x (x + 1) (x - 1)) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{4 \cdot 2}})$

Step 4.3.2

Multiply $4$ by $2$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} \frac{d}{d x} (x (x + 1) (x - 1)) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.4

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) \frac{d}{d x} (x - 1) + (x - 1) \frac{d}{d x} (x (x + 1))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.5

Differentiate.

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Step 4.5.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) (\frac{d}{d x} (x) + \frac{d}{d x} (- 1)) + (x - 1) \frac{d}{d x} (x (x + 1))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.5.2

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) (1 + \frac{d}{d x} (- 1)) + (x - 1) \frac{d}{d x} (x (x + 1))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.5.3

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) (1 + 0) + (x - 1) \frac{d}{d x} (x (x + 1))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.5.4

Simplify the expression.

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Step 4.5.4.1

Add $1$ and $0$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) \cdot 1 + (x - 1) \frac{d}{d x} (x (x + 1))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.5.4.2

Multiply $x$ by $1$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) \frac{d}{d x} (x (x + 1))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.6

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x \frac{d}{d x} (x + 1) + (x + 1) \frac{d}{d x} (x))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7

Differentiate.

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Step 4.7.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x (\frac{d}{d x} (x) + \frac{d}{d x} (1)) + (x + 1) \frac{d}{d x} (x))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.2

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x (1 + \frac{d}{d x} (1)) + (x + 1) \frac{d}{d x} (x))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.3

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x (1 + 0) + (x + 1) \frac{d}{d x} (x))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.4

Simplify the expression.

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Step 4.7.4.1

Add $1$ and $0$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x \cdot 1 + (x + 1) \frac{d}{d x} (x))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.4.2

Multiply $x$ by $1$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x + (x + 1) \frac{d}{d x} (x))) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.5

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x + (x + 1) \cdot 1)) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.6

Simplify by adding terms.

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Step 4.7.6.1

Multiply $x + 1$ by $1$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (x + x + 1)) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.7.6.2

Add $x$ and $x$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (2 x + 1)) - x (x + 1) (x - 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}})$

Step 4.8

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 $g (x) = x^{2} + 1$ .

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Step 4.8.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (2 x + 1)) - x (x + 1) (x - 1) (\frac{d}{d u} (u^{4}) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}})$

Step 4.8.2

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (2 x + 1)) - x (x + 1) (x - 1) (4 u^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}})$

Step 4.8.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (2 x + 1)) - x (x + 1) (x - 1) (4 {(x^{2} + 1)}^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}})$

Step 4.9

Simplify with factoring out.

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Step 4.9.1

Multiply $4$ by $- 1$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}})$

Step 4.9.2

Factor ${(x^{2} + 1)}^{3}$ out of ${(x^{2} + 1)}^{4} (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} [x^{2} + 1])$ .

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Step 4.9.2.1

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

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1))) - 4 x (x + 1) (x - 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}})$

Step 4.9.2.2

Factor ${(x^{2} + 1)}^{3}$ out of $- 4 x (x + 1) (x - 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} [x^{2} + 1])$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1))) + {(x^{2} + 1)}^{3} (- 4 x (x + 1) (x - 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{8}})$

Step 4.9.2.3

Factor ${(x^{2} + 1)}^{3}$ out of ${(x^{2} + 1)}^{3} ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1))) + {(x^{2} + 1)}^{3} (- 4 x (x + 1) (x - 1) (\frac{d}{d x} [x^{2} + 1]))$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{8}})$

Step 4.10

Cancel the common factors.

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Step 4.10.1

Factor ${(x^{2} + 1)}^{3}$ out of ${(x^{2} + 1)}^{8}$ .

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{3} {(x^{2} + 1)}^{5}})$

Step 4.10.2

Cancel the common factor.

$f^{4} (x) = 48 (\frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{3} {(x^{2} + 1)}^{5}})$

Step 4.10.3

Rewrite the expression.

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (\frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{5}})$

Step 4.11

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{5}})$

Step 4.12

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

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (2 x + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{5}})$

Step 4.13

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

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (2 x + 0)}{{(x^{2} + 1)}^{5}})$

Step 4.14

Simplify the expression.

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Step 4.14.1

Add $2 x$ and $0$ .

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 4 x (x + 1) (x - 1) (2 x)}{{(x^{2} + 1)}^{5}})$

Step 4.14.2

Multiply $2$ by $- 4$ .

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 x (x + 1) (x - 1) x}{{(x^{2} + 1)}^{5}})$

Step 4.15

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

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 (x \cdot x) (x + 1) (x - 1)}{{(x^{2} + 1)}^{5}})$

Step 4.16

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

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 (x \cdot x) (x + 1) (x - 1)}{{(x^{2} + 1)}^{5}})$

Step 4.17

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

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 x^{1 + 1} (x + 1) (x - 1)}{{(x^{2} + 1)}^{5}})$

Step 4.18

Add $1$ and $1$ .

$f^{4} (x) = 48 (\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 x^{2} (x + 1) (x - 1)}{{(x^{2} + 1)}^{5}})$

Step 4.19

Combine $48$ and $\frac{(x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 x^{2} (x + 1) (x - 1)}{{(x^{2} + 1)}^{5}}$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x (x + 1) + (x - 1) (2 x + 1)) - 8 x^{2} (x + 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20

Simplify.

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Step 4.20.1

Apply the distributive property.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x \cdot x + x \cdot 1 + (x - 1) (2 x + 1)) - 8 x^{2} (x + 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.2

Apply the distributive property.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x \cdot x + x \cdot 1 + (x - 1) (2 x + 1)) + (- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.3

Apply the distributive property.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x \cdot x + x \cdot 1 + (x - 1) (2 x + 1))) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4

Simplify the numerator.

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Step 4.20.4.1

Simplify each term.

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Step 4.20.4.1.1

Simplify each term.

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Step 4.20.4.1.1.1

Multiply $x$ by $x$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x \cdot 1 + (x - 1) (2 x + 1))) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.2

Multiply $x$ by $1$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + (x - 1) (2 x + 1))) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.3

Expand $(x - 1) (2 x + 1)$ using the FOIL Method.

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Step 4.20.4.1.1.3.1

Apply the distributive property.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + x (2 x + 1) - 1 (2 x + 1))) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.3.2

Apply the distributive property.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + x (2 x) + x \cdot 1 - 1 (2 x + 1))) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.3.3

Apply the distributive property.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4

Simplify and combine like terms.

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Step 4.20.4.1.1.4.1

Simplify each term.

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Step 4.20.4.1.1.4.1.1

Rewrite using the commutative property of multiplication.

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 x \cdot x + x \cdot 1 - 1 (2 x) - 1 \cdot 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4.1.2

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

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Step 4.20.4.1.1.4.1.2.1

Move $x$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 (x \cdot x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4.1.2.2

Multiply $x$ by $x$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 x^{2} + x \cdot 1 - 1 (2 x) - 1 \cdot 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4.1.3

Multiply $x$ by $1$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 x^{2} + x - 1 (2 x) - 1 \cdot 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4.1.4

Multiply $2$ by $- 1$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 x^{2} + x - 2 x - 1 \cdot 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4.1.5

Multiply $- 1$ by $1$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 x^{2} + x - 2 x - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.1.4.2

Subtract $2 x$ from $x$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + x + 2 x^{2} - x - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.2

Combine the opposite terms in $x^{2} + x + 2 x^{2} - x - 1$ .

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Step 4.20.4.1.2.1

Subtract $x$ from $x$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + 2 x^{2} + 0 - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.2.2

Add $x^{2} + 2 x^{2}$ and $0$ .

$f^{4} (x) = \frac{48 ((x^{2} + 1) (x^{2} + 2 x^{2} - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.3

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

$f^{4} (x) = \frac{48 ((x^{2} + 1) (3 x^{2} - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.4

Expand $(x^{2} + 1) (3 x^{2} - 1)$ using the FOIL Method.

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Step 4.20.4.1.4.1

Apply the distributive property.

$f^{4} (x) = \frac{48 (x^{2} (3 x^{2} - 1) + 1 (3 x^{2} - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.4.2

Apply the distributive property.

$f^{4} (x) = \frac{48 (x^{2} (3 x^{2}) + x^{2} \cdot - 1 + 1 (3 x^{2} - 1)) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.4.3

Apply the distributive property.

$f^{4} (x) = \frac{48 (x^{2} (3 x^{2}) + x^{2} \cdot - 1 + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5

Simplify and combine like terms.

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Step 4.20.4.1.5.1

Simplify each term.

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Step 4.20.4.1.5.1.1

Rewrite using the commutative property of multiplication.

$f^{4} (x) = \frac{48 (3 x^{2} x^{2} + x^{2} \cdot - 1 + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.2

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

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Step 4.20.4.1.5.1.2.1

Move $x^{2}$ .

$f^{4} (x) = \frac{48 (3 (x^{2} x^{2}) + x^{2} \cdot - 1 + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.2.2

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

$f^{4} (x) = \frac{48 (3 x^{2 + 2} + x^{2} \cdot - 1 + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.2.3

Add $2$ and $2$ .

$f^{4} (x) = \frac{48 (3 x^{4} + x^{2} \cdot - 1 + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.3

Move $- 1$ to the left of $x^{2}$ .

$f^{4} (x) = \frac{48 (3 x^{4} - 1 \cdot x^{2} + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f^{4} (x) = \frac{48 (3 x^{4} - x^{2} + 1 (3 x^{2}) + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.5

Multiply $3 x^{2}$ by $1$ .

$f^{4} (x) = \frac{48 (3 x^{4} - x^{2} + 3 x^{2} + 1 \cdot - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.1.6

Multiply $- 1$ by $1$ .

$f^{4} (x) = \frac{48 (3 x^{4} - x^{2} + 3 x^{2} - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.5.2

Add $- x^{2}$ and $3 x^{2}$ .

$f^{4} (x) = \frac{48 (3 x^{4} + 2 x^{2} - 1) + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.6

Apply the distributive property.

$f^{4} (x) = \frac{48 (3 x^{4}) + 48 (2 x^{2}) + 48 \cdot - 1 + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.7

Simplify.

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Step 4.20.4.1.7.1

Multiply $3$ by $48$ .

$f^{4} (x) = \frac{144 x^{4} + 48 (2 x^{2}) + 48 \cdot - 1 + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.7.2

Multiply $2$ by $48$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} + 48 \cdot - 1 + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.7.3

Multiply $48$ by $- 1$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 ((- 8 x^{2} x - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.8

Simplify each term.

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Step 4.20.4.1.8.1

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

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Step 4.20.4.1.8.1.1

Move $x$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 ((- 8 (x \cdot x^{2}) - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.8.1.2

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

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Step 4.20.4.1.8.1.2.1

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 ((- 8 (x \cdot x^{2}) - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.8.1.2.2

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 ((- 8 x^{1 + 2} - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.8.1.3

Add $1$ and $2$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 ((- 8 x^{3} - 8 x^{2} \cdot 1) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.8.2

Multiply $- 8$ by $1$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 ((- 8 x^{3} - 8 x^{2}) (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.9

Expand $(- 8 x^{3} - 8 x^{2}) (x - 1)$ using the FOIL Method.

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Step 4.20.4.1.9.1

Apply the distributive property.

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{3} (x - 1) - 8 x^{2} (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.9.2

Apply the distributive property.

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{3} x - 8 x^{3} \cdot - 1 - 8 x^{2} (x - 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.9.3

Apply the distributive property.

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{3} x - 8 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10

Simplify and combine like terms.

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Step 4.20.4.1.10.1

Simplify each term.

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Step 4.20.4.1.10.1.1

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

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Step 4.20.4.1.10.1.1.1

Move $x$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 (x \cdot x^{3}) - 8 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.1.2

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

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Step 4.20.4.1.10.1.1.2.1

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 (x \cdot x^{3}) - 8 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.1.2.2

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{1 + 3} - 8 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.1.3

Add $1$ and $3$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} - 8 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.2

Multiply $- 1$ by $- 8$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{3} - 8 x^{2} x - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.3

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

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Step 4.20.4.1.10.1.3.1

Move $x$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{3} - 8 (x \cdot x^{2}) - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.3.2

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

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Step 4.20.4.1.10.1.3.2.1

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{3} - 8 (x \cdot x^{2}) - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.3.2.2

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{3} - 8 x^{1 + 2} - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.3.3

Add $1$ and $2$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{3} - 8 x^{3} - 8 x^{2} \cdot - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.1.4

Multiply $- 1$ by $- 8$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{3} - 8 x^{3} + 8 x^{2})}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.2

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

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 0 + 8 x^{2})}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.10.3

Add $- 8 x^{4}$ and $0$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4} + 8 x^{2})}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.11

Apply the distributive property.

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 + 48 (- 8 x^{4}) + 48 (8 x^{2})}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.12

Multiply $- 8$ by $48$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 - 384 x^{4} + 48 (8 x^{2})}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.1.13

Multiply $8$ by $48$ .

$f^{4} (x) = \frac{144 x^{4} + 96 x^{2} - 48 - 384 x^{4} + 384 x^{2}}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.2

Subtract $384 x^{4}$ from $144 x^{4}$ .

$f^{4} (x) = \frac{- 240 x^{4} + 96 x^{2} - 48 + 384 x^{2}}{{(x^{2} + 1)}^{5}}$

Step 4.20.4.3

Add $96 x^{2}$ and $384 x^{2}$ .

$f^{4} (x) = \frac{- 240 x^{4} + 480 x^{2} - 48}{{(x^{2} + 1)}^{5}}$

Step 4.20.5

Factor $48$ out of $- 240 x^{4} + 480 x^{2} - 48$ .

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Step 4.20.5.1

Factor $48$ out of $- 240 x^{4}$ .

$f^{4} (x) = \frac{48 (- 5 x^{4}) + 480 x^{2} - 48}{{(x^{2} + 1)}^{5}}$

Step 4.20.5.2

Factor $48$ out of $480 x^{2}$ .

$f^{4} (x) = \frac{48 (- 5 x^{4}) + 48 (10 x^{2}) - 48}{{(x^{2} + 1)}^{5}}$

Step 4.20.5.3

Factor $48$ out of $- 48$ .

$f^{4} (x) = \frac{48 (- 5 x^{4}) + 48 (10 x^{2}) + 48 (- 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.5.4

Factor $48$ out of $48 (- 5 x^{4}) + 48 (10 x^{2})$ .

$f^{4} (x) = \frac{48 (- 5 x^{4} + 10 x^{2}) + 48 (- 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.5.5

Factor $48$ out of $48 (- 5 x^{4} + 10 x^{2}) + 48 (- 1)$ .

$f^{4} (x) = \frac{48 (- 5 x^{4} + 10 x^{2} - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.6

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

$f^{4} (x) = \frac{48 (- (5 x^{4}) + 10 x^{2} - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.7

Factor $- 1$ out of $10 x^{2}$ .

$f^{4} (x) = \frac{48 (- (5 x^{4}) - (- 10 x^{2}) - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.8

Factor $- 1$ out of $- (5 x^{4}) - (- 10 x^{2})$ .

$f^{4} (x) = \frac{48 (- (5 x^{4} - 10 x^{2}) - 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.9

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

$f^{4} (x) = \frac{48 (- (5 x^{4} - 10 x^{2}) - 1 \cdot 1)}{{(x^{2} + 1)}^{5}}$

Step 4.20.10

Factor $- 1$ out of $- (5 x^{4} - 10 x^{2}) - 1 (1)$ .

$f^{4} (x) = \frac{48 (- (5 x^{4} - 10 x^{2} + 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.11

Rewrite $- (5 x^{4} - 10 x^{2} + 1)$ as $- 1 (5 x^{4} - 10 x^{2} + 1)$ .

$f^{4} (x) = \frac{48 (- 1 (5 x^{4} - 10 x^{2} + 1))}{{(x^{2} + 1)}^{5}}$

Step 4.20.12

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{48 (5 x^{4} - 10 x^{2} + 1)}{{(x^{2} + 1)}^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{48 (5 x^{4} - 10 x^{2} + 1)}{{(x^{2} + 1)}^{5}}$ .

$- \frac{48 (5 x^{4} - 10 x^{2} + 1)}{{(x^{2} + 1)}^{5}}$