Calculus Examples

$y = \sqrt[5]{\frac{x}{x^{2} - 1}}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{x}{x^{2} - 1}}$ as ${(\frac{x}{x^{2} - 1})}^{\frac{1}{5}}$ .

$\frac{d}{d x} [{(\frac{x}{x^{2} - 1})}^{\frac{1}{5}}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = \frac{x}{x^{2} - 1}$ .

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Step 1.2.1

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

$\frac{d}{d u} [u^{\frac{1}{5}}] \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{5}$ .

$\frac{1}{5} u^{\frac{1}{5} - 1} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.2.3

Replace all occurrences of $u$ with $\frac{x}{x^{2} - 1}$ .

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1}{5} - 1} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1 - 1 \cdot 5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $5$ .

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{1 - 5}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.6.2

Subtract $5$ from $1$ .

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{\frac{- 4}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.7

Move the negative in front of the fraction.

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{d}{d x} [\frac{x}{x^{2} - 1}]$

Step 1.8

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

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{(x^{2} - 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 1.9

Differentiate.

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Step 1.9.1

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

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{(x^{2} - 1) \cdot 1 - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 1.9.2

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

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 1.9.3

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

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

Step 1.9.4

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

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - x (2 x + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

Step 1.9.5

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

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

Step 1.9.6

Simplify the expression.

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Step 1.9.6.1

Add $2 x$ and $0$ .

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

Step 1.9.6.2

Multiply $2$ by $- 1$ .

$\frac{1}{5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}} \frac{x^{2} - 1 - 2 x \cdot x}{{(x^{2} - 1)}^{2}}$

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Add $1$ and $1$ .

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

Step 1.14

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

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

Step 1.15

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

$\frac{- x^{2} - 1}{{(x^{2} - 1)}^{2} \cdot 5} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}}$

Step 1.16

Move $5$ to the left of ${(x^{2} - 1)}^{2}$ .

$\frac{- x^{2} - 1}{5 \cdot {(x^{2} - 1)}^{2}} {(\frac{x}{x^{2} - 1})}^{- \frac{4}{5}}$

Step 1.17

Simplify.

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Step 1.17.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 1.17.2

Apply the product rule to $\frac{x^{2} - 1}{x}$ .

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{2}} \cdot \frac{{(x^{2} - 1)}^{\frac{4}{5}}}{x^{\frac{4}{5}}}$

Step 1.17.3

Combine terms.

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Step 1.17.3.1

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

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

Step 1.17.3.2

Move ${(x^{2} - 1)}^{\frac{4}{5}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 1.17.3.3

Multiply ${(x^{2} - 1)}^{2}$ by ${(x^{2} - 1)}^{- \frac{4}{5}}$ by adding the exponents.

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Step 1.17.3.3.1

Move ${(x^{2} - 1)}^{- \frac{4}{5}}$ .

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

Step 1.17.3.3.2

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

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

Step 1.17.3.3.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.17.3.3.4

Combine $2$ and $\frac{5}{5}$ .

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

Step 1.17.3.3.5

Combine the numerators over the common denominator.

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

Step 1.17.3.3.6

Simplify the numerator.

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Step 1.17.3.3.6.1

Multiply $2$ by $5$ .

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

Step 1.17.3.3.6.2

Add $- 4$ and $10$ .

$\frac{- x^{2} - 1}{5 {(x^{2} - 1)}^{\frac{6}{5}} x^{\frac{4}{5}}}$

Step 1.17.4

Reorder terms.

$\frac{- x^{2} - 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

Step 1.17.5

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

$\frac{- (x^{2}) - 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

Step 1.17.6

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

$\frac{- (x^{2}) - 1 (1)}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$

Step 1.17.7

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

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

Step 1.17.8

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

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

Step 1.17.9

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

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Step 2.1

Since $- \frac{1}{5}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2} + 1}{5 x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}$ with respect to $x$ is $- \frac{1}{5} \frac{d}{d x} [\frac{x^{2} + 1}{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}]$ .

$- \frac{1}{5} \frac{d}{d x} [\frac{x^{2} + 1}{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}}]$

Step 2.2

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{2} + 1] - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

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

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.3.2

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

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (2 x + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.3.3

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

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.3.4

Add $2 x$ and $0$ .

$- \frac{1}{5} \cdot \frac{x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (2 x) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.4

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

$- \frac{1}{5} \cdot \frac{x^{1} x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.5

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

$- \frac{1}{5} \cdot \frac{x^{1 + \frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Write $1$ as a fraction with a common denominator.

$- \frac{1}{5} \cdot \frac{x^{\frac{5}{5} + \frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.6.2

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{x^{\frac{5 + 4}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.6.3

Add $5$ and $4$ .

$- \frac{1}{5} \cdot \frac{x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.6.4

Move $2$ to the left of $x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{1}{5} \cdot \frac{2 \cdot (x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}) - (x^{2} + 1) \frac{d}{d x} [x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}]}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.7

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 $f (x) = x^{\frac{4}{5}}$ and $g (x) = {(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} \frac{d}{d x} [{(x^{2} - 1)}^{\frac{6}{5}}] + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{6}{5}}$ and $g (x) = x^{2} - 1$ .

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Step 2.8.1

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{d}{d u} [u^{\frac{6}{5}}] \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.8.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{6}{5}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} u^{\frac{6}{5} - 1} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.8.3

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6}{5} - 1} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.10

Combine $- 1$ and $\frac{5}{5}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.11

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6 - 1 \cdot 5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.12

Simplify the numerator.

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Step 2.12.1

Multiply $- 1$ by $5$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{6 - 5}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.12.2

Subtract $5$ from $6$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6}{5} {(x^{2} - 1)}^{\frac{1}{5}} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.13

Combine fractions.

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Step 2.13.1

Combine $\frac{6}{5}$ and ${(x^{2} - 1)}^{\frac{1}{5}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (x^{\frac{4}{5}} (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}}}{5} \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.13.2

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} \frac{d}{d x} [x^{2} - 1] + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.14

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.15

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (2 x + \frac{d}{d x} [- 1]) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.16

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (2 x + 0) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.17

Combine fractions.

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Step 2.17.1

Add $2 x$ and $0$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}}{5} (2 x) + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.17.2

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{2 (6 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}})}{5} x + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.17.3

Multiply $6$ by $2$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}})}{5} x + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.17.4

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{4}{5}}) x}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.18

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} (x^{1} x^{\frac{4}{5}}))}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.19

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{1 + \frac{4}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.20

Simplify the expression.

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Step 2.20.1

Write $1$ as a fraction with a common denominator.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{5}{5} + \frac{4}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.20.2

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{5 + 4}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.20.3

Add $5$ and $4$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 ({(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}})}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{d}{d x} [x^{\frac{4}{5}}])}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.21

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.22

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.23

Combine $- 1$ and $\frac{5}{5}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.24

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.25

Simplify the numerator.

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Step 2.25.1

Multiply $- 1$ by $5$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{4 - 5}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.25.2

Subtract $5$ from $4$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{\frac{- 1}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.26

Move the negative in front of the fraction.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} (\frac{4}{5} x^{- \frac{1}{5}}))}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.27

Combine $\frac{4}{5}$ and $x^{- \frac{1}{5}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + {(x^{2} - 1)}^{\frac{6}{5}} \frac{4 x^{- \frac{1}{5}}}{5})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.28

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + \frac{{(x^{2} - 1)}^{\frac{6}{5}} (4 x^{- \frac{1}{5}})}{5})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.29

Simplify the expression.

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Step 2.29.1

Move $4$ to the left of ${(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + \frac{4 \cdot {(x^{2} - 1)}^{\frac{6}{5}} x^{- \frac{1}{5}}}{5})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.29.2

Move $x^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} + \frac{4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.30

To write $\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{5}}}{x^{\frac{1}{5}}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5} \cdot \frac{x^{\frac{1}{5}}}{x^{\frac{1}{5}}} + \frac{4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.31

Multiply $\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}}}{5}$ by $\frac{x^{\frac{1}{5}}}{x^{\frac{1}{5}}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) (\frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}} x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}} + \frac{4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}})}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.32

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{9}{5}} x^{\frac{1}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.33

Multiply $x^{\frac{9}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 2.33.1

Move $x^{\frac{1}{5}}$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} (x^{\frac{1}{5}} x^{\frac{9}{5}}) + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.33.2

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

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{1}{5} + \frac{9}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.33.3

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{1 + 9}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.33.4

Add $1$ and $9$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{\frac{10}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.33.5

Divide $10$ by $5$ .

$- \frac{1}{5} \cdot \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.34

Multiply $\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$ by $\frac{1}{5}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{{(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2} \cdot 5}$

Step 2.35

Move $5$ to the left of ${(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 \cdot {(x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}})}^{2}}$

Step 2.36

Simplify.

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Step 2.36.1

Apply the product rule to $x^{\frac{4}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} - (x^{2} + 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.2

Apply the distributive property.

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3

Simplify the numerator.

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Step 2.36.3.1

Simplify the numerator.

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Step 2.36.3.1.1

Factor $4 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $12 {(x^{2} - 1)}^{\frac{1}{5}} x^{2} + 4 {(x^{2} - 1)}^{\frac{6}{5}}$ .

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Step 2.36.3.1.1.1

Move ${(x^{2} - 1)}^{\frac{1}{5}}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{12 x^{2} {(x^{2} - 1)}^{\frac{1}{5}} + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.1.2

Factor $4 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $12 x^{2} {(x^{2} - 1)}^{\frac{1}{5}}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2}) + 4 {(x^{2} - 1)}^{\frac{6}{5}}}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.1.3

Factor $4 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $4 {(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2}) + 4 {(x^{2} - 1)}^{\frac{1}{5}} ({(x^{2} - 1)}^{\frac{5}{5}})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.1.4

Factor $4 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2}) + 4 {(x^{2} - 1)}^{\frac{1}{5}} ({(x^{2} - 1)}^{\frac{5}{5}})$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2} + {(x^{2} - 1)}^{\frac{5}{5}})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.2

Divide $5$ by $5$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2} + {(x^{2} - 1)}^{1})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.3

Simplify.

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (3 x^{2} + x^{2} - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.4

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

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (4 x^{2} - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.5

Rewrite $4 x^{2} - 1$ in a factored form.

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Step 2.36.3.1.5.1

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

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} ({(2 x)}^{2} - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.5.2

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

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} ({(2 x)}^{2} - 1^{2})}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.1.5.3

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

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} ((2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1 \cdot 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.2

Multiply $- 1$ by $1$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + (- x^{2} - 1) \frac{4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.3

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

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + \frac{(- x^{2} - 1) (4 {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.4

Move $4$ to the left of $- x^{2} - 1$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + \frac{4 \cdot (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.5

To write $2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ as a fraction with a common denominator, multiply by $\frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$ .

$- \frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} \cdot \frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}} + \frac{4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.6

Combine $2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ and $\frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$ .

$- \frac{\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}})}{5 x^{\frac{1}{5}}} + \frac{4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.7

Combine the numerators over the common denominator.

$- \frac{\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8

Rewrite $\frac{2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}$ in a factored form.

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Step 2.36.3.8.1

Factor $2 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $2 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{6}{5}} (5 x^{\frac{1}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)$ .

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Step 2.36.3.8.1.1

Reorder the expression.

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Step 2.36.3.8.1.1.1

Move ${(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{\frac{2 x^{\frac{9}{5}} \cdot 5 x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.1.1.2

Move $x^{\frac{9}{5}}$ .

$- \frac{\frac{2 \cdot 5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{6}{5}} + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.1.2

Factor $2 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $2 \cdot 5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{6}{5}}$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}}) + 4 (- x^{2} - 1) {(x^{2} - 1)}^{\frac{1}{5}} (2 x + 1) (2 x - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.1.3

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}}) + 2 {(x^{2} - 1)}^{\frac{1}{5}} ((2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.1.4

Factor $2 {(x^{2} - 1)}^{\frac{1}{5}}$ out of $2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}}) + 2 {(x^{2} - 1)}^{\frac{1}{5}} ((2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{9}{5}} x^{\frac{1}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.2

Multiply $x^{\frac{9}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 2.36.3.8.2.1

Move $x^{\frac{1}{5}}$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 (x^{\frac{1}{5}} x^{\frac{9}{5}}) {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.2.2

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{1}{5} + \frac{9}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.2.3

Combine the numerators over the common denominator.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{1 + 9}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.2.4

Add $1$ and $9$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{\frac{10}{5}} {(x^{2} - 1)}^{\frac{5}{5}} + (2 (- x^{2} - 1) (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.8.2.5

Divide $10$ by $5$ .

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

Step 2.36.3.9

Cancel the common factor of $5$ .

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Step 2.36.3.9.1

Cancel the common factor.

Step 2.36.3.9.2

Rewrite the expression.

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

Step 2.36.3.10

Simplify the numerator.

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Step 2.36.3.10.1

Simplify.

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

Step 2.36.3.10.2

Apply the distributive property.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2} x^{2} + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.3

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

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Step 2.36.3.10.3.1

Move $x^{2}$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 (x^{2} x^{2}) + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.3.2

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{2 + 2} + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.3.3

Add $2$ and $2$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} + 5 x^{2} \cdot - 1 + 2 (- x^{2} - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.4

Multiply $- 1$ by $5$ .

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

Step 2.36.3.10.5

Apply the distributive property.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (2 (- x^{2}) + 2 \cdot - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.6

Multiply $- 1$ by $2$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} + 2 \cdot - 1) (2 x + 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.7

Multiply $2$ by $- 1$ .

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

Step 2.36.3.10.8

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

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Step 2.36.3.10.8.1

Apply the distributive property.

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

Step 2.36.3.10.8.2

Apply the distributive property.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} (2 x) - 2 x^{2} \cdot 1 - 2 (2 x + 1)) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.8.3

Apply the distributive property.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 x^{2} (2 x) - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.9

Simplify each term.

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Step 2.36.3.10.9.1

Rewrite using the commutative property of multiplication.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 x^{2} x - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.9.2

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

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Step 2.36.3.10.9.2.1

Move $x$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 (x \cdot x^{2}) - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.9.2.2

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

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Step 2.36.3.10.9.2.2.1

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 (x^{1} x^{2}) - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.9.2.2.2

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} + (- 2 \cdot 2 x^{1 + 2} - 2 x^{2} \cdot 1 - 2 (2 x) - 2 \cdot 1) (2 x - 1))}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.9.2.3

Add $1$ and $2$ .

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

Step 2.36.3.10.9.3

Multiply $- 2$ by $2$ .

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

Step 2.36.3.10.9.4

Multiply $- 2$ by $1$ .

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

Step 2.36.3.10.9.5

Multiply $2$ by $- 2$ .

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

Step 2.36.3.10.9.6

Multiply $- 2$ by $1$ .

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

Step 2.36.3.10.10

Expand $(- 4 x^{3} - 2 x^{2} - 4 x - 2) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 x^{3} (2 x) - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.11

Simplify each term.

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Step 2.36.3.10.11.1

Rewrite using the commutative property of multiplication.

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 x^{3} x - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.11.2

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

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Step 2.36.3.10.11.2.1

Move $x$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 (x \cdot x^{3}) - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.11.2.2

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

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Step 2.36.3.10.11.2.2.1

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 (x^{1} x^{3}) - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.11.2.2.2

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

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

Step 2.36.3.10.11.2.3

Add $1$ and $3$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 4 \cdot 2 x^{4} - 4 x^{3} \cdot - 1 - 2 x^{2} (2 x) - 2 x^{2} \cdot - 1 - 4 x (2 x) - 4 x \cdot - 1 - 2 (2 x) - 2 \cdot - 1)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.11.3

Multiply $- 4$ by $2$ .

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

Step 2.36.3.10.11.4

Multiply $- 1$ by $- 4$ .

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

Step 2.36.3.10.11.5

Rewrite using the commutative property of multiplication.

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

Step 2.36.3.10.11.6

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

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Step 2.36.3.10.11.6.1

Move $x$ .

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

Step 2.36.3.10.11.6.2

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

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Step 2.36.3.10.11.6.2.1

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

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

Step 2.36.3.10.11.6.2.2

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

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

Step 2.36.3.10.11.6.3

Add $1$ and $2$ .

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

Step 2.36.3.10.11.7

Multiply $- 2$ by $2$ .

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

Step 2.36.3.10.11.8

Multiply $- 1$ by $- 2$ .

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

Step 2.36.3.10.11.9

Rewrite using the commutative property of multiplication.

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

Step 2.36.3.10.11.10

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

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Step 2.36.3.10.11.10.1

Move $x$ .

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

Step 2.36.3.10.11.10.2

Multiply $x$ by $x$ .

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

Step 2.36.3.10.11.11

Multiply $- 4$ by $2$ .

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

Step 2.36.3.10.11.12

Multiply $- 1$ by $- 4$ .

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

Step 2.36.3.10.11.13

Multiply $2$ by $- 2$ .

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

Step 2.36.3.10.11.14

Multiply $- 2$ by $- 1$ .

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

Step 2.36.3.10.12

Combine the opposite terms in $- 8 x^{4} + 4 x^{3} - 4 x^{3} + 2 x^{2} - 8 x^{2} + 4 x - 4 x + 2$ .

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Step 2.36.3.10.12.1

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 0 + 2 x^{2} - 8 x^{2} + 4 x - 4 x + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.12.2

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

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

Step 2.36.3.10.12.3

Subtract $4 x$ from $4 x$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (5 x^{4} - 5 x^{2} - 8 x^{4} + 2 x^{2} - 8 x^{2} + 0 + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.3.10.12.4

Add $- 8 x^{4} + 2 x^{2} - 8 x^{2}$ and $0$ .

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

Step 2.36.3.10.13

Subtract $8 x^{2}$ from $2 x^{2}$ .

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

Step 2.36.3.10.14

Subtract $8 x^{4}$ from $5 x^{4}$ .

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

Step 2.36.3.10.15

Subtract $6 x^{2}$ from $- 5 x^{2}$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 ({(x^{\frac{4}{5}})}^{2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.4

Combine terms.

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Step 2.36.4.1

Multiply the exponents in ${(x^{\frac{4}{5}})}^{2}$ .

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Step 2.36.4.1.1

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{4}{5} \cdot 2} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.4.1.2

Multiply $\frac{4}{5} \cdot 2$ .

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Step 2.36.4.1.2.1

Combine $\frac{4}{5}$ and $2$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{4 \cdot 2}{5}} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.4.1.2.2

Multiply $4$ by $2$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {({(x^{2} - 1)}^{\frac{6}{5}})}^{2})}$

Step 2.36.4.2

Multiply the exponents in ${({(x^{2} - 1)}^{\frac{6}{5}})}^{2}$ .

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Step 2.36.4.2.1

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

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{6}{5} \cdot 2})}$

Step 2.36.4.2.2

Multiply $\frac{6}{5} \cdot 2$ .

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Step 2.36.4.2.2.1

Combine $\frac{6}{5}$ and $2$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{6 \cdot 2}{5}})}$

Step 2.36.4.2.2.2

Multiply $6$ by $2$ .

$- \frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

Step 2.36.4.3

Rewrite $\frac{\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}}{5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$ as a product.

$- (\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}} \cdot \frac{1}{5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}}})$

Step 2.36.4.4

Multiply $\frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}}}$ by $\frac{1}{5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$ .

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{5 x^{\frac{1}{5}} (5 x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

Step 2.36.4.5

Multiply $5$ by $5$ .

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{1}{5}} (x^{\frac{8}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

Step 2.36.4.6

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

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{8}{5} + \frac{1}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$

Step 2.36.4.7

Combine the numerators over the common denominator.

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{8 + 1}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$

Step 2.36.4.8

Add $8$ and $1$ .

$- \frac{2 {(x^{2} - 1)}^{\frac{1}{5}} (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{12}{5}}}$

Step 2.36.4.9

Move ${(x^{2} - 1)}^{\frac{1}{5}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{12}{5}} {(x^{2} - 1)}^{- \frac{1}{5}}}$

Step 2.36.4.10

Multiply ${(x^{2} - 1)}^{\frac{12}{5}}$ by ${(x^{2} - 1)}^{- \frac{1}{5}}$ by adding the exponents.

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Step 2.36.4.10.1

Move ${(x^{2} - 1)}^{- \frac{1}{5}}$ .

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} ({(x^{2} - 1)}^{- \frac{1}{5}} {(x^{2} - 1)}^{\frac{12}{5}})}$

Step 2.36.4.10.2

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

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{- \frac{1}{5} + \frac{12}{5}}}$

Step 2.36.4.10.3

Combine the numerators over the common denominator.

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{- 1 + 12}{5}}}$

Step 2.36.4.10.4

Add $- 1$ and $12$ .

$- \frac{2 (- 3 x^{4} - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.5

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

$- \frac{2 (- (3 x^{4}) - 11 x^{2} + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.6

Factor $- 1$ out of $- 11 x^{2}$ .

$- \frac{2 (- (3 x^{4}) - (11 x^{2}) + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.7

Factor $- 1$ out of $- (3 x^{4}) - (11 x^{2})$ .

$- \frac{2 (- (3 x^{4} + 11 x^{2}) + 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.8

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

$- \frac{2 (- (3 x^{4} + 11 x^{2}) - 1 (- 2))}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.9

Factor $- 1$ out of $- (3 x^{4} + 11 x^{2}) - 1 (- 2)$ .

$- \frac{2 (- (3 x^{4} + 11 x^{2} - 2))}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.10

Rewrite $- (3 x^{4} + 11 x^{2} - 2)$ as $- 1 (3 x^{4} + 11 x^{2} - 2)$ .

$- \frac{2 (- 1 (3 x^{4} + 11 x^{2} - 2))}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.11

Move the negative in front of the fraction.

$- - \frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.12

Multiply $- 1$ by $- 1$ .

$1 \frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$

Step 2.36.13

Multiply $\frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$ by $1$ .

$f'' (x) = \frac{2 (3 x^{4} + 11 x^{2} - 2)}{25 x^{\frac{9}{5}} {(x^{2} - 1)}^{\frac{11}{5}}}$