Calculus Examples

$y = \ln (\sqrt{\frac{x - 1}{x + 1}})$

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${(\frac{x - 1}{x + 1})}^{\frac{1}{2}}$ .

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

Step 2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with ${(\frac{x - 1}{x + 1})}^{\frac{1}{2}}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{2}$ with $\frac{x - 1}{x + 1}$ .

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

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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 $f (x) = x - 1$ and $g (x) = x + 1$ .

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

Step 10

Differentiate.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Simplify the expression.

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Step 10.4.1

Add $1$ and $0$ .

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

Step 10.4.2

Multiply $x + 1$ by $1$ .

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

Step 10.5

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

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

Step 10.6

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

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

Step 10.7

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

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

Step 10.8

Combine fractions.

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Step 10.8.1

Add $1$ and $0$ .

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

Step 10.8.2

Multiply $- 1$ by $1$ .

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

Step 10.8.3

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

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

Step 10.8.4

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

Apply the distributive property.

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

Step 11.5

Combine terms.

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Step 11.5.1

Multiply by the reciprocal of the fraction to divide by $\frac{{(x - 1)}^{\frac{1}{2}}}{{(x + 1)}^{\frac{1}{2}}}$ .

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

Step 11.5.2

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

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

Step 11.5.3

Multiply $- 1$ by $- 1$ .

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

Step 11.5.4

Subtract $x$ from $x$ .

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

Step 11.5.5

Add $0$ and $1$ .

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

Step 11.5.6

Add $1$ and $1$ .

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

Step 11.5.7

Cancel the common factor of $2$ .

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Step 11.5.7.1

Cancel the common factor.

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

Step 11.5.7.2

Rewrite the expression.

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

Step 11.5.8

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

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

Step 11.5.9

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

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

Step 11.5.10

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

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Step 11.5.10.1

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

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

Step 11.5.10.2

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

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

Step 11.5.10.3

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

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

Step 11.5.10.4

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

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

Step 11.5.10.5

Combine the numerators over the common denominator.

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

Step 11.5.10.6

Simplify the numerator.

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Step 11.5.10.6.1

Multiply $2$ by $2$ .

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

Step 11.5.10.6.2

Add $- 1$ and $4$ .

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

Step 11.5.11

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

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

Step 11.5.12

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

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Step 11.5.12.1

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

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

Step 11.5.12.2

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

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

Step 11.5.12.3

Combine the numerators over the common denominator.

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

Step 11.5.12.4

Add $1$ and $1$ .

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

Step 11.5.12.5

Divide $2$ by $2$ .

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

Step 11.5.13

Simplify ${(x - 1)}^{1} {(x + 1)}^{\frac{3}{2}}$ .

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

Step 11.5.14

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

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

Step 11.5.15

Simplify the denominator.

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Step 11.5.15.1

Multiply ${(x + 1)}^{\frac{3}{2}}$ by ${(x + 1)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.5.15.1.1

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

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

Step 11.5.15.1.2

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

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

Step 11.5.15.1.3

Combine the numerators over the common denominator.

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

Step 11.5.15.1.4

Add $- 1$ and $3$ .

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

Step 11.5.15.1.5

Divide $2$ by $2$ .

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

Step 11.5.15.2

Simplify $(x - 1) {(x + 1)}^{1}$ .

$\frac{1}{(x - 1) (x + 1)}$

Step 11.6

Reorder terms.

$\frac{1}{(x + 1) (x - 1)}$