Calculus Examples

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

Step 1

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

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

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 $g (x) = {(\frac{1 + \tan (x)}{1 - \tan (x)})}^{\frac{1}{2}}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

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 $g (x) = \frac{1 + \tan (x)}{1 - \tan (x)}$ .

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Step 3.1

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

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

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 $n = \frac{1}{2}$ .

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

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

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

Step 10

Differentiate.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Add $0$ and $\frac{d}{d x} [\tan (x)]$ .

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

Step 11

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 12

Differentiate.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

Add $0$ and $\frac{d}{d x} [- \tan (x)]$ .

Step 12.4

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

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

Step 12.5

Multiply.

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Step 12.5.1

Multiply $- 1$ by $- 1$ .

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

Step 12.5.2

Multiply $1 + \tan (x)$ by $1$ .

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

Step 13

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

Step 14

Combine fractions.

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Step 14.1

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

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

Step 14.2

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

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

Step 15

Simplify.

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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

Step 15.4

Apply the distributive property.

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

Step 15.5

Apply the distributive property.

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

Step 15.6

Combine terms.

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Step 15.6.1

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

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

Step 15.6.2

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

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

Step 15.6.3

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

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

Step 15.6.4

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

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

Step 15.6.5

Add $\sec^{2} (x)$ and $\sec^{2} (x)$ .

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

Step 15.6.6

Add $- \tan (x) \sec^{2} (x)$ and $\tan (x) \sec^{2} (x)$ .

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

Step 15.6.7

Add $0$ and $2 \sec^{2} (x)$ .

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

Step 15.6.8

Cancel the common factor of $2$ .

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Step 15.6.8.1

Cancel the common factor.

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

Step 15.6.8.2

Rewrite the expression.

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

Step 15.6.9

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

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

Step 15.6.10

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

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

Step 15.6.11

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

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Step 15.6.11.1

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

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

Step 15.6.11.2

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

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

Step 15.6.11.3

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

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

Step 15.6.11.4

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

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

Step 15.6.11.5

Combine the numerators over the common denominator.

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

Step 15.6.11.6

Simplify the numerator.

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Step 15.6.11.6.1

Multiply $2$ by $2$ .

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

Step 15.6.11.6.2

Add $- 1$ and $4$ .

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

Step 15.6.12

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

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

Step 15.6.13

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

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Step 15.6.13.1

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

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

Step 15.6.13.2

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

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

Step 15.6.13.3

Combine the numerators over the common denominator.

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

Step 15.6.13.4

Add $1$ and $1$ .

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

Step 15.6.13.5

Divide $2$ by $2$ .

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

Step 15.6.14

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

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

Step 15.6.15

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

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

Step 15.6.16

Simplify the denominator.

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Step 15.6.16.1

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

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Step 15.6.16.1.1

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

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

Step 15.6.16.1.2

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

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

Step 15.6.16.1.3

Combine the numerators over the common denominator.

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

Step 15.6.16.1.4

Add $- 1$ and $3$ .

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

Step 15.6.16.1.5

Divide $2$ by $2$ .

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

Step 15.6.16.2

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

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