Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\arcsin (u)$ with respect to $u$ is $\frac{1}{\sqrt{1 - u^{2}}}$ .

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

Step 1.3

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$\frac{1}{\sqrt{1 - {(\frac{x^{2}}{x^{2} - 1})}^{2}}} \cdot \frac{2 (x^{2} - 1) x - x^{2} (2 x + 0)}{{(x^{2} - 1)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2 x$ and $0$ .

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

Step 3.6.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $2$ .

$\frac{1}{\sqrt{1 - {(\frac{x^{2}}{x^{2} - 1})}^{2}}} \cdot \frac{2 (x^{2} - 1) x - 2 x^{3}}{{(x^{2} - 1)}^{2}}$

Step 7

Multiply $\frac{1}{\sqrt{1 - {(\frac{x^{2}}{x^{2} - 1})}^{2}}}$ by $\frac{2 (x^{2} - 1) x - 2 x^{3}}{{(x^{2} - 1)}^{2}}$ .

$\frac{2 (x^{2} - 1) x - 2 x^{3}}{\sqrt{1 - {(\frac{x^{2}}{x^{2} - 1})}^{2}} {(x^{2} - 1)}^{2}}$

Step 8

Simplify.

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Step 8.1

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

$\frac{2 (x^{2} - 1) x - 2 x^{3}}{\sqrt{1 - \frac{{(x^{2})}^{2}}{{(x^{2} - 1)}^{2}}} {(x^{2} - 1)}^{2}}$

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Simplify the numerator.

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Step 8.4.1

Simplify each term.

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Step 8.4.1.1

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

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Step 8.4.1.1.1

Move $x$ .

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

Step 8.4.1.1.2

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

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Step 8.4.1.1.2.1

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

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

Step 8.4.1.1.2.2

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

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

Step 8.4.1.1.3

Add $1$ and $2$ .

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

Step 8.4.1.2

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} - 2 x - 2 x^{3}}{\sqrt{1 - \frac{{(x^{2})}^{2}}{{(x^{2} - 1)}^{2}}} {(x^{2} - 1)}^{2}}$

Step 8.4.2

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

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Step 8.4.2.1

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

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

Step 8.4.2.2

Add $- 2 x$ and $0$ .

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

Step 8.5

Combine terms.

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Step 8.5.1

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

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Step 8.5.1.1

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

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

Step 8.5.1.2

Multiply $2$ by $2$ .

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

Step 8.5.2

Move the negative in front of the fraction.

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

Step 8.6

Simplify the denominator.

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

Apply the product rule to $(x + 1) (x - 1)$ .

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

Step 8.7

Simplify the denominator.

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Step 8.7.1

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

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

Step 8.7.2

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

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

Step 8.7.3

Apply the product rule to $(x + 1) (x - 1)$ .

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

Step 8.7.4

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

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

Step 8.7.5

Combine the numerators over the common denominator.

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

Step 8.7.6

Reorder terms.

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

Step 8.7.7

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

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Step 8.7.7.1

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

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

Step 8.7.7.2

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

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

Step 8.7.7.3

Reorder $- {(x^{2})}^{2}$ and ${((x + 1) (x - 1))}^{2}$ .

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

Step 8.7.7.4

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

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

Step 8.7.7.5

Simplify.

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Step 8.7.7.5.1

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

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Step 8.7.7.5.1.1

Apply the distributive property.

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

Step 8.7.7.5.1.2

Apply the distributive property.

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

Step 8.7.7.5.1.3

Apply the distributive property.

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

Step 8.7.7.5.2

Simplify and combine like terms.

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Step 8.7.7.5.2.1

Simplify each term.

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Step 8.7.7.5.2.1.1

Multiply $x$ by $x$ .

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

Step 8.7.7.5.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 8.7.7.5.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 8.7.7.5.2.1.4

Multiply $x$ by $1$ .

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

Step 8.7.7.5.2.1.5

Multiply $- 1$ by $1$ .

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

Step 8.7.7.5.2.2

Add $- x$ and $x$ .

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

Step 8.7.7.5.2.3

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

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

Step 8.7.7.5.3

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

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

Step 8.7.7.5.4

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

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Step 8.7.7.5.4.1

Apply the distributive property.

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

Step 8.7.7.5.4.2

Apply the distributive property.

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

Step 8.7.7.5.4.3

Apply the distributive property.

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

Step 8.7.7.5.5

Simplify and combine like terms.

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Step 8.7.7.5.5.1

Simplify each term.

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Step 8.7.7.5.5.1.1

Multiply $x$ by $x$ .

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

Step 8.7.7.5.5.1.2

Move $- 1$ to the left of $x$ .

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

Step 8.7.7.5.5.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 8.7.7.5.5.1.4

Multiply $x$ by $1$ .

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

Step 8.7.7.5.5.1.5

Multiply $- 1$ by $1$ .

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

Step 8.7.7.5.5.2

Add $- x$ and $x$ .

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

Step 8.7.7.5.5.3

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

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

Step 8.7.7.5.6

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

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

Step 8.7.7.5.7

Subtract $1$ from $0$ .

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

Step 8.7.8

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

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

Step 8.7.9

Move the negative in front of the fraction.

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

Step 8.7.10

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

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Step 8.7.10.1

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

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

Step 8.7.10.2

Factor the perfect power ${((x + 1) (x - 1))}^{2}$ out of ${(x + 1)}^{2} {(x - 1)}^{2}$ .

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

Step 8.7.10.3

Rearrange the fraction $\frac{1^{2} (2 x^{2} - 1)}{{((x + 1) (x - 1))}^{2} \cdot 1}$ .

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

Step 8.7.10.4

Reorder $- 1$ and ${(\frac{1}{(x + 1) (x - 1)})}^{2}$ .

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

Step 8.7.10.5

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

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

Step 8.7.10.6

Add parentheses.

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

Step 8.7.11

Pull terms out from under the radical.

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

Step 8.7.12

One to any power is one.

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

Step 8.7.13

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

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

Step 8.7.14

Combine exponents.

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Step 8.7.14.1

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

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

Step 8.7.14.2

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

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

Step 8.7.15

Reduce the expression $\frac{\sqrt{- (2 x^{2} - 1)} {(x + 1)}^{2} {(x - 1)}^{2}}{(x + 1) (x - 1)}$ by cancelling the common factors.

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Step 8.7.15.1

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

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

Step 8.7.15.2

Cancel the common factor.

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

Step 8.7.15.3

Rewrite the expression.

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

Step 8.7.16

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

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Step 8.7.16.1

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

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

Step 8.7.16.2

Cancel the common factors.

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Step 8.7.16.2.1

Multiply by $1$ .

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

Step 8.7.16.2.2

Cancel the common factor.

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

Step 8.7.16.2.3

Rewrite the expression.

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

Step 8.7.16.2.4

Divide $\sqrt{- (2 x^{2} - 1)} (x + 1) (x - 1)$ by $1$ .

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

Step 8.7.17

Apply the distributive property.

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

Step 8.7.18

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

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

Step 8.7.19

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

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Step 8.7.19.1

Apply the distributive property.

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

Step 8.7.19.2

Apply the distributive property.

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

Step 8.7.19.3

Apply the distributive property.

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

Step 8.7.20

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

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Step 8.7.20.1

Reorder the factors in the terms $\sqrt{- (2 x^{2} - 1)} x \cdot - 1$ and $\sqrt{- (2 x^{2} - 1)} x$ .

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

Step 8.7.20.2

Add $- x \sqrt{- (2 x^{2} - 1)}$ and $x \sqrt{- (2 x^{2} - 1)}$ .

$- \frac{2 x}{\sqrt{- (2 x^{2} - 1)} x \cdot x + 0 + \sqrt{- (2 x^{2} - 1)} \cdot - 1}$

Step 8.7.20.3

Add $\sqrt{- (2 x^{2} - 1)} x \cdot x$ and $0$ .

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

Step 8.7.21

Simplify each term.

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Step 8.7.21.1

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

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Step 8.7.21.1.1

Move $x$ .

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

Step 8.7.21.1.2

Multiply $x$ by $x$ .

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

Step 8.7.21.2

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

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

Step 8.7.21.3

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

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

Step 8.7.22

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

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Step 8.7.22.1

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

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

Step 8.7.22.2

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

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

Step 8.7.22.3

Factor $\sqrt{- (2 x^{2} - 1)}$ out of $\sqrt{- (2 x^{2} - 1)} (x^{2}) + \sqrt{- (2 x^{2} - 1)} \cdot - 1$ .

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

Step 8.7.23

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

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

Step 8.7.24

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

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

Step 8.8

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

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

Step 8.9

Combine and simplify the denominator.

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Step 8.9.1

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

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

Step 8.9.2

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

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

Step 8.9.3

Raise $\sqrt{- (2 x^{2} - 1)}$ to the power of $1$ .

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

Step 8.9.4

Raise $\sqrt{- (2 x^{2} - 1)}$ to the power of $1$ .

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

Step 8.9.5

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

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

Step 8.9.6

Add $1$ and $1$ .

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

Step 8.9.7

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

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Step 8.9.7.1

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

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

Step 8.9.7.2

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

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

Step 8.9.7.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 8.9.7.4

Cancel the common factor of $2$ .

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Step 8.9.7.4.1

Cancel the common factor.

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

Step 8.9.7.4.2

Rewrite the expression.

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

Step 8.9.7.5

Simplify.

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

Step 8.10

Simplify the denominator.

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Step 8.10.1

Rewrite.

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

Step 8.10.2

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

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

Step 8.10.3

Raise $\sqrt{- (2 x^{2} - 1)}$ to the power of $1$ .

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

Step 8.10.4

Raise $\sqrt{- (2 x^{2} - 1)}$ to the power of $1$ .

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

Step 8.10.5

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

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

Step 8.10.6

Add $1$ and $1$ .

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

Step 8.10.7

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

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Step 8.10.7.1

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

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

Step 8.10.7.2

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

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

Step 8.10.7.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 8.10.7.4

Cancel the common factor of $2$ .

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Step 8.10.7.4.1

Cancel the common factor.

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

Step 8.10.7.4.2

Rewrite the expression.

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

Step 8.10.7.5

Simplify.

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

Step 8.10.8

Remove unnecessary parentheses.

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

Step 8.10.9

Factor out negative.

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

Step 8.11

Move the negative in front of the fraction.

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

Step 8.12

Multiply $- - \frac{2 x \sqrt{- (2 x^{2} - 1)}}{(x + 1) (x - 1) (2 x^{2} - 1)}$ .

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Step 8.12.1

Multiply $- 1$ by $- 1$ .

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

Step 8.12.2

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

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