Calculus Examples

$y = \frac{x}{x^{2} + 1}$

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Add $1$ and $1$ .

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

Step 1.7

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

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

Step 2

Find the second derivative.

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Step 2.1

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Multiply $2$ by $2$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $2$ by $- 1$ .

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

Step 2.2.6

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

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

Step 2.2.7

Add $- 2 x$ and $0$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Differentiate.

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Step 2.4.1

Multiply $2$ by $- 1$ .

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Simplify the expression.

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Step 2.4.5.1

Add $2 x$ and $0$ .

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

Step 2.4.5.2

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

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

Step 2.4.5.3

Multiply $2$ by $- 2$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Simplify the numerator.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.5.3.1.2

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

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

Step 2.5.3.1.3

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

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Step 2.5.3.1.3.1

Apply the distributive property.

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

Step 2.5.3.1.3.2

Apply the distributive property.

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

Step 2.5.3.1.3.3

Apply the distributive property.

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

Step 2.5.3.1.4

Simplify and combine like terms.

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Step 2.5.3.1.4.1

Simplify each term.

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Step 2.5.3.1.4.1.1

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

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Step 2.5.3.1.4.1.1.1

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

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

Step 2.5.3.1.4.1.1.2

Add $2$ and $2$ .

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

Step 2.5.3.1.4.1.2

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

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

Step 2.5.3.1.4.1.3

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

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

Step 2.5.3.1.4.1.4

Multiply $1$ by $1$ .

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

Step 2.5.3.1.4.2

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

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

Step 2.5.3.1.5

Apply the distributive property.

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

Step 2.5.3.1.6

Simplify.

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Step 2.5.3.1.6.1

Multiply $2$ by $- 2$ .

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

Step 2.5.3.1.6.2

Multiply $- 2$ by $1$ .

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

Step 2.5.3.1.7

Apply the distributive property.

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

Step 2.5.3.1.8

Simplify.

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Step 2.5.3.1.8.1

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

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Step 2.5.3.1.8.1.1

Move $x$ .

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

Step 2.5.3.1.8.1.2

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

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Step 2.5.3.1.8.1.2.1

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

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

Step 2.5.3.1.8.1.2.2

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

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

Step 2.5.3.1.8.1.3

Add $1$ and $4$ .

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

Step 2.5.3.1.8.2

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

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Step 2.5.3.1.8.2.1

Move $x$ .

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

Step 2.5.3.1.8.2.2

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

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Step 2.5.3.1.8.2.2.1

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

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

Step 2.5.3.1.8.2.2.2

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

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

Step 2.5.3.1.8.2.3

Add $1$ and $2$ .

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

Step 2.5.3.1.9

Simplify each term.

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Step 2.5.3.1.9.1

Multiply $- 1$ by $- 4$ .

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

Step 2.5.3.1.9.2

Multiply $- 4$ by $1$ .

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

Step 2.5.3.1.10

Simplify each term.

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Step 2.5.3.1.10.1

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

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Step 2.5.3.1.10.1.1

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

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Step 2.5.3.1.10.1.1.1

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

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

Step 2.5.3.1.10.1.1.2

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

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

Step 2.5.3.1.10.1.2

Add $2$ and $1$ .

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

Step 2.5.3.1.10.2

Multiply $x$ by $1$ .

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

Step 2.5.3.1.11

Expand $(4 x^{2} - 4) (x^{3} + x)$ using the FOIL Method.

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Step 2.5.3.1.11.1

Apply the distributive property.

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

Step 2.5.3.1.11.2

Apply the distributive property.

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

Step 2.5.3.1.11.3

Apply the distributive property.

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

Step 2.5.3.1.12

Simplify and combine like terms.

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Step 2.5.3.1.12.1

Simplify each term.

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Step 2.5.3.1.12.1.1

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

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Step 2.5.3.1.12.1.1.1

Move $x^{3}$ .

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

Step 2.5.3.1.12.1.1.2

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

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

Step 2.5.3.1.12.1.1.3

Add $3$ and $2$ .

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

Step 2.5.3.1.12.1.2

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

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Step 2.5.3.1.12.1.2.1

Move $x$ .

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

Step 2.5.3.1.12.1.2.2

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

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Step 2.5.3.1.12.1.2.2.1

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

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

Step 2.5.3.1.12.1.2.2.2

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

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

Step 2.5.3.1.12.1.2.3

Add $1$ and $2$ .

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

Step 2.5.3.1.12.2

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

$f'' (x) = \frac{- 2 x^{5} - 4 x^{3} - 2 x + 4 x^{5} + 0 - 4 x}{{(x^{2} + 1)}^{4}}$

Step 2.5.3.1.12.3

Add $4 x^{5}$ and $0$ .

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

Step 2.5.3.2

Add $- 2 x^{5}$ and $4 x^{5}$ .

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

Step 2.5.3.3

Subtract $4 x$ from $- 2 x$ .

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

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Factor $2 x$ out of $2 x^{5} - 4 x^{3} - 6 x$ .

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Step 2.5.4.1.1

Factor $2 x$ out of $2 x^{5}$ .

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

Step 2.5.4.1.2

Factor $2 x$ out of $- 4 x^{3}$ .

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

Step 2.5.4.1.3

Factor $2 x$ out of $- 6 x$ .

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

Step 2.5.4.1.4

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

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

Step 2.5.4.1.5

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

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

Step 2.5.4.2

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

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

Step 2.5.4.3

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

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

Step 2.5.4.4

Factor $u^{2} - 2 u - 3$ using the AC method.

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Step 2.5.4.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 2.5.4.4.2

Write the factored form using these integers.

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

Step 2.5.4.5

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

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

Step 2.5.5

Cancel the common factor of $x^{2} + 1$ and ${(x^{2} + 1)}^{4}$ .

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Step 2.5.5.1

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

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

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

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

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

Step 2.5.5.2.2

Cancel the common factor.

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

Step 2.5.5.2.3

Rewrite the expression.

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $3$ by $2$ .

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

Step 3.4

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 $f (x) = x$ and $g (x) = x^{2} - 3$ .

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

Step 3.5

Differentiate.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.5.4

Add $2 x$ and $0$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Differentiate using the Power Rule.

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Step 3.9.1

Add $1$ and $1$ .

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

Step 3.9.2

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

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

Step 3.9.3

Simplify by adding terms.

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Step 3.9.3.1

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

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

Step 3.9.3.2

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

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

Step 3.10

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 3.10.1

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

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

Step 3.10.2

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

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

Step 3.10.3

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

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

Step 3.11

Simplify with factoring out.

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Step 3.11.1

Multiply $3$ by $- 1$ .

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

Step 3.11.2

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

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Step 3.11.2.1

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

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

Step 3.11.2.2

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

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

Step 3.11.2.3

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

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

Step 3.12

Cancel the common factors.

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Step 3.12.1

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

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

Step 3.12.2

Cancel the common factor.

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

Step 3.12.3

Rewrite the expression.

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

Simplify the expression.

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Step 3.16.1

Add $2 x$ and $0$ .

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

Step 3.16.2

Multiply $2$ by $- 3$ .

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

Add $1$ and $1$ .

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

Step 3.21

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

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

Step 3.22

Simplify.

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Step 3.22.1

Apply the distributive property.

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

Step 3.22.2

Apply the distributive property.

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

Step 3.22.3

Simplify the numerator.

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Step 3.22.3.1

Simplify each term.

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Step 3.22.3.1.1

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

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Step 3.22.3.1.1.1

Apply the distributive property.

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

Step 3.22.3.1.1.2

Apply the distributive property.

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

Step 3.22.3.1.1.3

Apply the distributive property.

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

Step 3.22.3.1.2

Simplify and combine like terms.

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Step 3.22.3.1.2.1

Simplify each term.

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Step 3.22.3.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.22.3.1.2.1.2

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

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Step 3.22.3.1.2.1.2.1

Move $x^{2}$ .

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

Step 3.22.3.1.2.1.2.2

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

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

Step 3.22.3.1.2.1.2.3

Add $2$ and $2$ .

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

Step 3.22.3.1.2.1.3

Move $- 3$ to the left of $x^{2}$ .

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

Step 3.22.3.1.2.1.4

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

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

Step 3.22.3.1.2.1.5

Multiply $- 3$ by $1$ .

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

Step 3.22.3.1.2.2

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

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

Step 3.22.3.1.2.3

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

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

Step 3.22.3.1.3

Apply the distributive property.

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

Step 3.22.3.1.4

Multiply $3$ by $2$ .

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

Step 3.22.3.1.5

Multiply $2$ by $- 3$ .

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

Step 3.22.3.1.6

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

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Step 3.22.3.1.6.1

Move $x^{2}$ .

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

Step 3.22.3.1.6.2

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

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

Step 3.22.3.1.6.3

Add $2$ and $2$ .

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

Step 3.22.3.1.7

Multiply $- 6$ by $2$ .

$f''' (x) = \frac{6 x^{4} - 6 - 12 x^{4} + 2 (- 6 x^{2} \cdot - 3)}{{(x^{2} + 1)}^{4}}$

Step 3.22.3.1.8

Multiply $- 3$ by $- 6$ .

$f''' (x) = \frac{6 x^{4} - 6 - 12 x^{4} + 2 (18 x^{2})}{{(x^{2} + 1)}^{4}}$

Step 3.22.3.1.9

Multiply $18$ by $2$ .

$f''' (x) = \frac{6 x^{4} - 6 - 12 x^{4} + 36 x^{2}}{{(x^{2} + 1)}^{4}}$

Step 3.22.3.2

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

$f''' (x) = \frac{- 6 x^{4} - 6 + 36 x^{2}}{{(x^{2} + 1)}^{4}}$

Step 3.22.4

Reorder terms.

$f''' (x) = \frac{- 6 x^{4} + 36 x^{2} - 6}{{(x^{2} + 1)}^{4}}$

Step 3.22.5

Factor $6$ out of $- 6 x^{4} + 36 x^{2} - 6$ .

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Step 3.22.5.1

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

$f''' (x) = \frac{6 (- x^{4}) + 36 x^{2} - 6}{{(x^{2} + 1)}^{4}}$

Step 3.22.5.2

Factor $6$ out of $36 x^{2}$ .

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

Step 3.22.5.3

Factor $6$ out of $- 6$ .

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

Step 3.22.5.4

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

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

Step 3.22.5.5

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

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

Step 3.22.6

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

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

Step 3.22.7

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

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

Step 3.22.8

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

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

Step 3.22.9

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

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

Step 3.22.10

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

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

Step 3.22.11

Rewrite $- (x^{4} - 6 x^{2} + 1)$ as $- 1 (x^{4} - 6 x^{2} + 1)$ .

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

Step 3.22.12

Move the negative in front of the fraction.

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

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6 (x^{4} - 6 x^{2} + 1)}{{(x^{2} + 1)}^{4}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{x^{4} - 6 x^{2} + 1}{{(x^{2} + 1)}^{4}}]$ .

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

Step 4.2

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

Step 4.3

Differentiate.

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

Multiply $4$ by $2$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} \frac{d}{d x} (x^{4} - 6 x^{2} + 1) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.2

By the Sum Rule, the derivative of $x^{4} - 6 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (1)) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.3

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (1)) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.4

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (1)) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.5

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 6 (2 x) + \frac{d}{d x} (1)) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.6

Multiply $2$ by $- 6$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x + \frac{d}{d x} (1)) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.7

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x + 0) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

Step 4.3.8

Add $4 x^{3} - 12 x$ and $0$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x) - (x^{4} - 6 x^{2} + 1) \frac{d}{d x} {(x^{2} + 1)}^{4}}{{(x^{2} + 1)}^{8}}$

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

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Step 4.4.1

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x) - (x^{4} - 6 x^{2} + 1) (\frac{d}{d u} (u^{4}) \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}}$

Step 4.4.2

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x) - (x^{4} - 6 x^{2} + 1) (4 u^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}}$

Step 4.4.3

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x) - (x^{4} - 6 x^{2} + 1) (4 {(x^{2} + 1)}^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}}$

Step 4.5

Simplify with factoring out.

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Step 4.5.1

Multiply $4$ by $- 1$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{4} (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}}$

Step 4.5.2

Factor ${(x^{2} + 1)}^{3}$ out of ${(x^{2} + 1)}^{4} (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} [x^{2} + 1])$ .

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Step 4.5.2.1

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (4 x^{3} - 12 x)) - 4 (x^{4} - 6 x^{2} + 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{8}}$

Step 4.5.2.2

Factor ${(x^{2} + 1)}^{3}$ out of $- 4 (x^{4} - 6 x^{2} + 1) ({(x^{2} + 1)}^{3} \frac{d}{d x} [x^{2} + 1])$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (4 x^{3} - 12 x)) + {(x^{2} + 1)}^{3} (- 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{8}}$

Step 4.5.2.3

Factor ${(x^{2} + 1)}^{3}$ out of ${(x^{2} + 1)}^{3} ((x^{2} + 1) (4 x^{3} - 12 x)) + {(x^{2} + 1)}^{3} (- 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} [x^{2} + 1]))$ .

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{8}}$

Step 4.6

Cancel the common factors.

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Step 4.6.1

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

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{3} {(x^{2} + 1)}^{5}}$

Step 4.6.2

Cancel the common factor.

$f^{4} (x) = - 6 \frac{{(x^{2} + 1)}^{3} ((x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} (x^{2} + 1)))}{{(x^{2} + 1)}^{3} {(x^{2} + 1)}^{5}}$

Step 4.6.3

Rewrite the expression.

$f^{4} (x) = - 6 \frac{(x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} (x^{2} + 1))}{{(x^{2} + 1)}^{5}}$

Step 4.7

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

$f^{4} (x) = - 6 \frac{(x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{5}}$

Step 4.8

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

$f^{4} (x) = - 6 \frac{(x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (2 x + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{5}}$

Step 4.9

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

$f^{4} (x) = - 6 \frac{(x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (2 x + 0)}{{(x^{2} + 1)}^{5}}$

Step 4.10

Combine fractions.

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Step 4.10.1

Add $2 x$ and $0$ .

$f^{4} (x) = - 6 \frac{(x^{2} + 1) (4 x^{3} - 12 x) - 4 (x^{4} - 6 x^{2} + 1) (2 x)}{{(x^{2} + 1)}^{5}}$

Step 4.10.2

Multiply $2$ by $- 4$ .

$f^{4} (x) = - 6 \frac{(x^{2} + 1) (4 x^{3} - 12 x) - 8 (x^{4} - 6 x^{2} + 1) x}{{(x^{2} + 1)}^{5}}$

Step 4.10.3

Combine $- 6$ and $\frac{(x^{2} + 1) (4 x^{3} - 12 x) - 8 (x^{4} - 6 x^{2} + 1) x}{{(x^{2} + 1)}^{5}}$ .

$f^{4} (x) = \frac{- 6 ((x^{2} + 1) (4 x^{3} - 12 x) - 8 (x^{4} - 6 x^{2} + 1) x)}{{(x^{2} + 1)}^{5}}$

Step 4.10.4

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{(6) ((x^{2} + 1) (4 x^{3} - 12 x) - 8 (x^{4} - 6 x^{2} + 1) x)}{{(x^{2} + 1)}^{5}}$

Step 4.11

Simplify.

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Step 4.11.1

Apply the distributive property.

$f^{4} (x) = - \frac{6 ((x^{2} + 1) (4 x^{3} - 12 x) + (- 8 x^{4} - 8 (- 6 x^{2}) - 8 \cdot 1) x)}{{(x^{2} + 1)}^{5}}$

Step 4.11.2

Apply the distributive property.

$f^{4} (x) = - \frac{6 ((x^{2} + 1) (4 x^{3} - 12 x) - 8 x^{4} x - 8 (- 6 x^{2}) x - 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.3

Apply the distributive property.

$f^{4} (x) = - \frac{6 ((x^{2} + 1) (4 x^{3} - 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

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 $(x^{2} + 1) (4 x^{3} - 12 x)$ using the FOIL Method.

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Step 4.11.4.1.1.1

Apply the distributive property.

$f^{4} (x) = - \frac{6 (x^{2} (4 x^{3} - 12 x) + 1 (4 x^{3} - 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.1.2

Apply the distributive property.

$f^{4} (x) = - \frac{6 (x^{2} (4 x^{3}) + x^{2} (- 12 x) + 1 (4 x^{3} - 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.1.3

Apply the distributive property.

$f^{4} (x) = - \frac{6 (x^{2} (4 x^{3}) + x^{2} (- 12 x) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2

Simplify and combine like terms.

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Step 4.11.4.1.2.1

Simplify each term.

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Step 4.11.4.1.2.1.1

Rewrite using the commutative property of multiplication.

$f^{4} (x) = - \frac{6 (4 x^{2} x^{3} + x^{2} (- 12 x) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.2

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

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Step 4.11.4.1.2.1.2.1

Move $x^{3}$ .

$f^{4} (x) = - \frac{6 (4 (x^{3} x^{2}) + x^{2} (- 12 x) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.2.2

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

$f^{4} (x) = - \frac{6 (4 x^{3 + 2} + x^{2} (- 12 x) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.2.3

Add $3$ and $2$ .

$f^{4} (x) = - \frac{6 (4 x^{5} + x^{2} (- 12 x) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.3

Rewrite using the commutative property of multiplication.

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 x^{2} x + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.4

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

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Step 4.11.4.1.2.1.4.1

Move $x$ .

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 (x \cdot x^{2}) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.4.2

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

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Step 4.11.4.1.2.1.4.2.1

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

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 (x \cdot x^{2}) + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.4.2.2

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

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 x^{1 + 2} + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.4.3

Add $1$ and $2$ .

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 x^{3} + 1 (4 x^{3}) + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.5

Multiply $4 x^{3}$ by $1$ .

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 x^{3} + 4 x^{3} + 1 (- 12 x)) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.1.6

Multiply $- 12 x$ by $1$ .

$f^{4} (x) = - \frac{6 (4 x^{5} - 12 x^{3} + 4 x^{3} - 12 x) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.2.2

Add $- 12 x^{3}$ and $4 x^{3}$ .

$f^{4} (x) = - \frac{6 (4 x^{5} - 8 x^{3} - 12 x) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.3

Apply the distributive property.

$f^{4} (x) = - \frac{6 (4 x^{5}) + 6 (- 8 x^{3}) + 6 (- 12 x) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.4

Simplify.

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Step 4.11.4.1.4.1

Multiply $4$ by $6$ .

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

Step 4.11.4.1.4.2

Multiply $- 8$ by $6$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} + 6 (- 12 x) + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.4.3

Multiply $- 12$ by $6$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x + 6 (- 8 x^{4} x) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.5

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

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Step 4.11.4.1.5.1

Move $x$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x + 6 (- 8 (x \cdot x^{4})) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.5.2

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

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Step 4.11.4.1.5.2.1

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

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x + 6 (- 8 (x \cdot x^{4})) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.5.2.2

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

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x + 6 (- 8 x^{1 + 4}) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.5.3

Add $1$ and $4$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x + 6 (- 8 x^{5}) + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.6

Multiply $- 8$ by $6$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 6 (- 8 (- 6 x^{2}) x) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.7

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

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Step 4.11.4.1.7.1

Move $x$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 6 (- 8 (- 6 (x \cdot x^{2}))) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.7.2

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

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Step 4.11.4.1.7.2.1

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

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 6 (- 8 (- 6 (x \cdot x^{2}))) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.7.2.2

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

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 6 (- 8 (- 6 x^{1 + 2})) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.7.3

Add $1$ and $2$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 6 (- 8 (- 6 x^{3})) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.8

Multiply $- 6$ by $- 8$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 6 (48 x^{3}) + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.9

Multiply $48$ by $6$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 288 x^{3} + 6 (- 8 \cdot (1 x))}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.10

Multiply $- 8$ by $1$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 288 x^{3} + 6 (- 8 x)}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.1.11

Multiply $- 8$ by $6$ .

$f^{4} (x) = - \frac{24 x^{5} - 48 x^{3} - 72 x - 48 x^{5} + 288 x^{3} - 48 x}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.2

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

$f^{4} (x) = - \frac{- 24 x^{5} - 48 x^{3} - 72 x + 288 x^{3} - 48 x}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.3

Add $- 48 x^{3}$ and $288 x^{3}$ .

$f^{4} (x) = - \frac{- 24 x^{5} + 240 x^{3} - 72 x - 48 x}{{(x^{2} + 1)}^{5}}$

Step 4.11.4.4

Subtract $48 x$ from $- 72 x$ .

$f^{4} (x) = - \frac{- 24 x^{5} + 240 x^{3} - 120 x}{{(x^{2} + 1)}^{5}}$

Step 4.11.5

Factor $24 x$ out of $- 24 x^{5} + 240 x^{3} - 120 x$ .

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Step 4.11.5.1

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

$f^{4} (x) = - \frac{24 x (- x^{4}) + 240 x^{3} - 120 x}{{(x^{2} + 1)}^{5}}$

Step 4.11.5.2

Factor $24 x$ out of $240 x^{3}$ .

$f^{4} (x) = - \frac{24 x (- x^{4}) + 24 x (10 x^{2}) - 120 x}{{(x^{2} + 1)}^{5}}$

Step 4.11.5.3

Factor $24 x$ out of $- 120 x$ .

$f^{4} (x) = - \frac{24 x (- x^{4}) + 24 x (10 x^{2}) + 24 x (- 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.5.4

Factor $24 x$ out of $24 x (- x^{4}) + 24 x (10 x^{2})$ .

$f^{4} (x) = - \frac{24 x (- x^{4} + 10 x^{2}) + 24 x (- 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.5.5

Factor $24 x$ out of $24 x (- x^{4} + 10 x^{2}) + 24 x (- 5)$ .

$f^{4} (x) = - \frac{24 x (- x^{4} + 10 x^{2} - 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.6

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

$f^{4} (x) = - \frac{24 x (- (x^{4}) + 10 x^{2} - 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.7

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

$f^{4} (x) = - \frac{24 x (- (x^{4}) - (- 10 x^{2}) - 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.8

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

$f^{4} (x) = - \frac{24 x (- (x^{4} - 10 x^{2}) - 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.9

Rewrite $- 5$ as $- 1 (5)$ .

$f^{4} (x) = - \frac{24 x (- (x^{4} - 10 x^{2}) - 1 \cdot 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.10

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

$f^{4} (x) = - \frac{24 x (- (x^{4} - 10 x^{2} + 5))}{{(x^{2} + 1)}^{5}}$

Step 4.11.11

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

$f^{4} (x) = - \frac{24 x (- 1 (x^{4} - 10 x^{2} + 5))}{{(x^{2} + 1)}^{5}}$

Step 4.11.12

Move the negative in front of the fraction.

$f^{4} (x) = \frac{(24 x) (x^{4} - 10 x^{2} + 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.13

Multiply $- 1$ by $- 1$ .

$f^{4} (x) = 1 (\frac{(24 x) (x^{4} - 10 x^{2} + 5)}{{(x^{2} + 1)}^{5}})$

Step 4.11.14

Multiply $\frac{(24 x) (x^{4} - 10 x^{2} + 5)}{{(x^{2} + 1)}^{5}}$ by $1$ .

$f^{4} (x) = \frac{(24 x) (x^{4} - 10 x^{2} + 5)}{{(x^{2} + 1)}^{5}}$

Step 4.11.15

Reorder factors in $\frac{(24 x) (x^{4} - 10 x^{2} + 5)}{{(x^{2} + 1)}^{5}}$ .

$f^{4} (x) = \frac{24 x (x^{4} - 10 x^{2} + 5)}{{(x^{2} + 1)}^{5}}$