Calculus Examples

$f (x) = 3 \ln (\sec (x) + \tan (x))$

Step 1

Find the first derivative.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \ln (\sec (x) + \tan (x))$ with respect to $x$ is $3 \frac{d}{d x} [\ln (\sec (x) + \tan (x))]$ .

$3 \frac{d}{d x} [\ln (\sec (x) + \tan (x))]$

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 $f (x) = \ln (x)$ and $g (x) = \sec (x) + \tan (x)$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\sec (x) + \tan (x)$ .

$3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x) + \tan (x)])$

Step 1.2.2

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

$3 (\frac{1}{u} \frac{d}{d x} [\sec (x) + \tan (x)])$

Step 1.2.3

Replace all occurrences of $u$ with $\sec (x) + \tan (x)$ .

$3 (\frac{1}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)])$

Step 1.3

Differentiate using the Sum Rule.

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Step 1.3.1

Combine $\frac{1}{\sec (x) + \tan (x)}$ and $3$ .

$\frac{3}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)]$

Step 1.3.2

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

$\frac{3}{\sec (x) + \tan (x)} (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])$

Step 1.4

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])$

Step 1.5

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

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

Step 1.6

Simplify.

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Step 1.6.1

Reorder the factors of $\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x))$ .

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Multiply $\sec (x) \tan (x) \frac{3}{\sec (x) + \tan (x)}$ .

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Step 1.6.3.1

Combine $\frac{3}{\sec (x) + \tan (x)}$ and $\sec (x)$ .

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

Step 1.6.3.2

Combine $\frac{3 \sec (x)}{\sec (x) + \tan (x)}$ and $\tan (x)$ .

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

Step 1.6.4

Combine $\sec^{2} (x)$ and $\frac{3}{\sec (x) + \tan (x)}$ .

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

Step 1.6.5

Move $3$ to the left of $\sec^{2} (x)$ .

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

Step 1.6.6

Combine the numerators over the common denominator.

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

Step 1.6.7

Factor $3 \sec (x)$ out of $3 \sec (x) \tan (x) + 3 \sec^{2} (x)$ .

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Step 1.6.7.1

Factor $3 \sec (x)$ out of $3 \sec (x) \tan (x)$ .

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

Step 1.6.7.2

Factor $3 \sec (x)$ out of $3 \sec^{2} (x)$ .

$\frac{3 \sec (x) (\tan (x)) + 3 \sec (x) (\sec (x))}{\sec (x) + \tan (x)}$

Step 1.6.7.3

Factor $3 \sec (x)$ out of $3 \sec (x) (\tan (x)) + 3 \sec (x) (\sec (x))$ .

$f' (x) = \frac{3 \sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}$

Step 2

Find the second derivative.

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Step 2.1

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

$3 \frac{d}{d x} [\frac{\sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}]$

Step 2.2

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

$3 \frac{(\sec (x) + \tan (x)) \frac{d}{d x} [\sec (x) (\tan (x) + \sec (x))] - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.3

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) \frac{d}{d x} [\tan (x) + \sec (x)] + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.4

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [\sec (x)]) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.5

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \frac{d}{d x} [\sec (x)]) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.6

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.7

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) (\sec (x) \tan (x))) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.8

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

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.9

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.10

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

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

Step 2.11

Combine $3$ and $\frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$ .

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

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

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

Step 2.12.2

Apply the distributive property.

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

Step 2.12.3

Apply the distributive property.

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

Step 2.12.4

Apply the distributive property.

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

Step 2.12.5

Apply the distributive property.

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

Step 2.12.6

Simplify the numerator.

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Step 2.12.6.1

Simplify each term.

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Step 2.12.6.1.1

Simplify each term.

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Step 2.12.6.1.1.1

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 2.12.6.1.1.1.1

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

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Step 2.12.6.1.1.1.1.1

Raise $\sec (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.1.1.2

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

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

Step 2.12.6.1.1.1.2

Add $1$ and $2$ .

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

Step 2.12.6.1.1.2

Multiply $\sec (x) (\sec (x) \tan (x))$ .

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Step 2.12.6.1.1.2.1

Raise $\sec (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.2.2

Raise $\sec (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.2.3

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

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

Step 2.12.6.1.1.2.4

Add $1$ and $1$ .

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

Step 2.12.6.1.1.3

Multiply $\tan (x) \sec (x) \tan (x)$ .

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Step 2.12.6.1.1.3.1

Raise $\tan (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.3.2

Raise $\tan (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.3.3

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

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

Step 2.12.6.1.1.3.4

Add $1$ and $1$ .

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

Step 2.12.6.1.1.4

Multiply $\sec (x) \sec (x)$ .

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Step 2.12.6.1.1.4.1

Raise $\sec (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.4.2

Raise $\sec (x)$ to the power of $1$ .

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

Step 2.12.6.1.1.4.3

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

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

Step 2.12.6.1.1.4.4

Add $1$ and $1$ .

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

Step 2.12.6.1.2

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

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

Step 2.12.6.1.3

Expand $(\sec (x) + \tan (x)) (\sec^{3} (x) + 2 \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2.12.6.1.4

Simplify each term.

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Step 2.12.6.1.4.1

Multiply $\sec (x)$ by $\sec^{3} (x)$ by adding the exponents.

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Step 2.12.6.1.4.1.1

Multiply $\sec (x)$ by $\sec^{3} (x)$ .

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Step 2.12.6.1.4.1.1.1

Raise $\sec (x)$ to the power of $1$ .

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

Step 2.12.6.1.4.1.1.2

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

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

Step 2.12.6.1.4.1.2

Add $1$ and $3$ .

$\frac{3 (\sec^{4} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.2

Rewrite using the commutative property of multiplication.

$\frac{3 (\sec^{4} (x) + 2 \sec (x) \sec^{2} (x) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.3

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 2.12.6.1.4.3.1

Move $\sec^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 (\sec^{2} (x) \sec (x)) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.3.2

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

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Step 2.12.6.1.4.3.2.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 (\sec^{4} (x) + 2 (\sec^{2} (x) \sec^{1} (x)) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.3.2.2

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

$\frac{3 (\sec^{4} (x) + 2 {\sec (x)}^{2 + 1} \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.3.3

Add $2$ and $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.4

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

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Step 2.12.6.1.4.4.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.4.2

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec^{1} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.4.3

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + {\sec (x)}^{1 + 1} \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.4.4

Add $1$ and $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.5

Rewrite using the commutative property of multiplication.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan (x) \sec^{2} (x) \tan (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.6

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

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Step 2.12.6.1.4.6.1

Raise $\tan (x)$ to the power of $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 (\tan^{1} (x) \tan (x)) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.6.2

Raise $\tan (x)$ to the power of $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 (\tan^{1} (x) \tan^{1} (x)) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.6.3

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 {\tan (x)}^{1 + 1} \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.6.4

Add $1$ and $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.7

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

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Step 2.12.6.1.4.7.1

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{2} (x) \tan (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.7.2

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

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Step 2.12.6.1.4.7.2.1

Raise $\tan (x)$ to the power of $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{2} (x) \tan^{1} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.7.2.2

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

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + {\tan (x)}^{2 + 1} \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.4.7.3

Add $2$ and $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.5

Reorder the factors of $\tan (x) \sec^{3} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.6

Add $2 \sec^{3} (x) \tan (x)$ and $\sec^{3} (x) \tan (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.7

Reorder the factors of $2 \tan^{2} (x) \sec^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{2} (x) \tan^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.8

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

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + 3 \sec^{2} (x) \tan^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.9

Apply the distributive property.

$\frac{3 \sec^{4} (x) + 3 (3 \sec^{3} (x) \tan (x)) + 3 (3 \sec^{2} (x) \tan^{2} (x)) + 3 (\tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.10

Simplify.

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Step 2.12.6.1.10.1

Multiply $3$ by $3$ .

$\frac{3 \sec^{4} (x) + 9 (\sec^{3} (x) \tan (x)) + 3 (3 \sec^{2} (x) \tan^{2} (x)) + 3 (\tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.10.2

Multiply $3$ by $3$ .

$\frac{3 \sec^{4} (x) + 9 (\sec^{3} (x) \tan (x)) + 9 (\sec^{2} (x) \tan^{2} (x)) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.11

Remove parentheses.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.12

Multiply $- \sec (x) \sec (x)$ .

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Step 2.12.6.1.12.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - (\sec^{1} (x) \sec (x))) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.12.2

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - (\sec^{1} (x) \sec^{1} (x))) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.12.3

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - {\sec (x)}^{1 + 1}) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.12.4

Add $1$ and $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.13

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

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Step 2.12.6.1.13.1

Apply the distributive property.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec (x) \tan (x) (\sec (x) \tan (x) + \sec^{2} (x)) - \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.13.2

Apply the distributive property.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec (x) \tan (x) (\sec (x) \tan (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.13.3

Apply the distributive property.

Step 2.12.6.1.14

Simplify and combine like terms.

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Step 2.12.6.1.14.1

Simplify each term.

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Step 2.12.6.1.14.1.1

Multiply $- \sec (x) \tan (x) (\sec (x) \tan (x))$ .

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Step 2.12.6.1.14.1.1.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- (\sec^{1} (x) \sec (x)) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.2

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- (\sec^{1} (x) \sec^{1} (x)) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.3

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- {\sec (x)}^{1 + 1} \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.4

Add $1$ and $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.5

Raise $\tan (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) (\tan^{1} (x) \tan (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.6

Raise $\tan (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) (\tan^{1} (x) \tan^{1} (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.7

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) {\tan (x)}^{1 + 1} - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.1.8

Add $1$ and $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.2

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 2.12.6.1.14.1.2.1

Move $\sec^{2} (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - (\sec^{2} (x) \sec (x)) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.2.2

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

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Step 2.12.6.1.14.1.2.2.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - (\sec^{2} (x) \sec^{1} (x)) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.2.2.2

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - {\sec (x)}^{2 + 1} \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.2.3

Add $2$ and $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.3

Multiply $\sec^{2} (x)$ by $\sec (x)$ by adding the exponents.

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Step 2.12.6.1.14.1.3.1

Move $\sec (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - (\sec (x) \sec^{2} (x)) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.3.2

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

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Step 2.12.6.1.14.1.3.2.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - (\sec^{1} (x) \sec^{2} (x)) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.3.2.2

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - {\sec (x)}^{1 + 2} \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.3.3

Add $1$ and $2$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.4

Multiply $\sec^{2} (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 2.12.6.1.14.1.4.1

Move $\sec^{2} (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - (\sec^{2} (x) \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.4.2

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

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - {\sec (x)}^{2 + 2})}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.14.1.4.3

Add $2$ and $2$ .

Step 2.12.6.1.14.2

Subtract $\sec^{3} (x) \tan (x)$ from $- \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x) - \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.15

Apply the distributive property.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x)) + 3 (- 2 \sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.16

Simplify.

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Step 2.12.6.1.16.1

Multiply $- 1$ by $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) + 3 (- 2 \sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.16.2

Multiply $- 2$ by $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) - 6 (\sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.16.3

Multiply $- 1$ by $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) - 6 (\sec^{3} (x) \tan (x)) - 3 \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.1.17

Remove parentheses.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) - 3 \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.2

Combine the opposite terms in $3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) - 3 \sec^{4} (x)$ .

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Step 2.12.6.2.1

Subtract $3 \sec^{4} (x)$ from $3 \sec^{4} (x)$ .

$\frac{9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) + 0}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.2.2

Add $9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x)$ and $0$ .

$\frac{9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x)}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.3

Subtract $6 \sec^{3} (x) \tan (x)$ from $9 \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.6.4

Subtract $3 \sec^{2} (x) \tan^{2} (x)$ from $9 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{3 \sec^{3} (x) \tan (x) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.7

Simplify the numerator.

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Step 2.12.7.1

Factor $3 \sec (x) \tan (x)$ out of $3 \sec^{3} (x) \tan (x) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)$ .

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Step 2.12.7.1.1

Factor $3 \sec (x) \tan (x)$ out of $3 \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Step 2.12.7.1.2

Factor $3 \sec (x) \tan (x)$ out of $6 \sec^{2} (x) \tan^{2} (x)$ .

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

Step 2.12.7.1.3

Factor $3 \sec (x) \tan (x)$ out of $3 \tan^{3} (x) \sec (x)$ .

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

Step 2.12.7.1.4

Factor $3 \sec (x) \tan (x)$ out of $3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x))$ .

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

Step 2.12.7.1.5

Factor $3 \sec (x) \tan (x)$ out of $3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))$ .

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

Step 2.12.7.2

Factor using the perfect square rule.

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Step 2.12.7.2.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sec (x) \tan (x) = 2 \cdot \sec (x) \cdot \tan (x)$

Step 2.12.7.2.2

Rewrite the polynomial.

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

Step 2.12.7.2.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \sec (x)$ and $b = \tan (x)$ .

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

Step 2.12.8

Cancel the common factor of ${(\sec (x) + \tan (x))}^{2}$ .

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Step 2.12.8.1

Cancel the common factor.

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

Step 2.12.8.2

Divide $3 \sec (x) \tan (x)$ by $1$ .

$f'' (x) = 3 \sec (x) \tan (x)$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $3 \sec (x) \tan (x)$ .

$3 \sec (x) \tan (x)$