Calculus Examples

$y = \ln (\sec (2 x) + \tan (2 x))$

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

By the Sum Rule, the derivative of $\sec (2 x) + \tan (2 x)$ with respect to $x$ is $\frac{d}{d x} [\sec (2 x)] + \frac{d}{d x} [\tan (2 x)]$ .

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

Step 1.3

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

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Step 1.3.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 1.4.2

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

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

Step 1.4.3

Simplify the expression.

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Step 1.4.3.1

Multiply $2$ by $1$ .

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

Step 1.4.3.2

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

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

Step 1.5

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

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Step 1.5.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

$\frac{1}{\sec (2 x) + \tan (2 x)} (2 \sec (2 x) \tan (2 x) + \frac{d}{d u_{3}} [\tan (u_{3})] \frac{d}{d x} [2 x])$

Step 1.5.2

The derivative of $\tan (u_{3})$ with respect to $u_{3}$ is $\sec^{2} (u_{3})$ .

$\frac{1}{\sec (2 x) + \tan (2 x)} (2 \sec (2 x) \tan (2 x) + \sec^{2} (u_{3}) \frac{d}{d x} [2 x])$

Step 1.5.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 1.6

Differentiate.

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Step 1.6.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 1.6.2

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

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

Step 1.6.3

Simplify the expression.

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Step 1.6.3.1

Multiply $2$ by $1$ .

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

Step 1.6.3.2

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

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

Step 1.7

Simplify.

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Step 1.7.1

Reorder the factors of $\frac{1}{\sec (2 x) + \tan (2 x)} (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x))$ .

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

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

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Step 1.7.3.1

Combine $\frac{1}{\sec (2 x) + \tan (2 x)}$ and $2$ .

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

Step 1.7.3.2

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

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

Step 1.7.3.3

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

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

Step 1.7.4

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

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Step 1.7.4.1

Combine $\frac{1}{\sec (2 x) + \tan (2 x)}$ and $2$ .

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

Step 1.7.4.2

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

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

Step 1.7.5

Combine the numerators over the common denominator.

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

Step 1.7.6

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

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Step 1.7.6.1

Factor $2 \sec (2 x)$ out of $2 \sec (2 x) \tan (2 x)$ .

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

Step 1.7.6.2

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

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

Step 1.7.6.3

Factor $2 \sec (2 x)$ out of $2 \sec (2 x) (\tan (2 x)) + 2 \sec (2 x) (\sec (2 x))$ .

$f' (x) = \frac{2 \sec (2 x) (\tan (2 x) + \sec (2 x))}{\sec (2 x) + \tan (2 x)}$

Step 2

Find the second derivative.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 \sec (2 x) (\tan (2 x) + \sec (2 x))}{\sec (2 x) + \tan (2 x)}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{\sec (2 x) (\tan (2 x) + \sec (2 x))}{\sec (2 x) + \tan (2 x)}]$ .

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

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 $f (x) = \sec (2 x) (\tan (2 x) + \sec (2 x))$ and $g (x) = \sec (2 x) + \tan (2 x)$ .

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

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 $f (x) = \sec (2 x)$ and $g (x) = \tan (2 x) + \sec (2 x)$ .

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

Step 2.4

By the Sum Rule, the derivative of $\tan (2 x) + \sec (2 x)$ with respect to $x$ is $\frac{d}{d x} [\tan (2 x)] + \frac{d}{d x} [\sec (2 x)]$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

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

Step 2.5.3

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 2.6

Differentiate.

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Step 2.6.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.6.2

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

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

Step 2.6.3

Simplify the expression.

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Step 2.6.3.1

Multiply $2$ by $1$ .

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

Step 2.6.3.2

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

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

Step 2.7

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

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Step 2.7.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 2.7.2

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

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

Step 2.7.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 2.8

Differentiate.

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Step 2.8.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.8.2

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

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

Step 2.8.3

Simplify the expression.

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Step 2.8.3.1

Multiply $2$ by $1$ .

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

Step 2.8.3.2

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

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

Step 2.9

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

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Step 2.9.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

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

Step 2.9.2

The derivative of $\sec (u_{3})$ with respect to $u_{3}$ is $\sec (u_{3}) \tan (u_{3})$ .

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

Step 2.9.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 2.10

Differentiate.

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Step 2.10.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.10.2

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

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

Step 2.10.3

Simplify the expression.

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Step 2.10.3.1

Multiply $2$ by $1$ .

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

Step 2.10.3.2

Move $2$ to the left of $(\tan (2 x) + \sec (2 x)) \sec (2 x) \tan (2 x)$ .

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

Step 2.10.4

By the Sum Rule, the derivative of $\sec (2 x) + \tan (2 x)$ with respect to $x$ is $\frac{d}{d x} [\sec (2 x)] + \frac{d}{d x} [\tan (2 x)]$ .

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

Step 2.11

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

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Step 2.11.1

To apply the Chain Rule, set $u_{4}$ as $2 x$ .

$2 \frac{(\sec (2 x) + \tan (2 x)) (\sec (2 x) (2 \sec^{2} (2 x) + 2 \sec (2 x) \tan (2 x)) + 2 (\tan (2 x) + \sec (2 x)) \sec (2 x) \tan (2 x)) - \sec (2 x) (\tan (2 x) + \sec (2 x)) (\frac{d}{d u_{4}} [\sec (u_{4})] \frac{d}{d x} [2 x] + \frac{d}{d x} [\tan (2 x)])}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.11.2

The derivative of $\sec (u_{4})$ with respect to $u_{4}$ is $\sec (u_{4}) \tan (u_{4})$ .

$2 \frac{(\sec (2 x) + \tan (2 x)) (\sec (2 x) (2 \sec^{2} (2 x) + 2 \sec (2 x) \tan (2 x)) + 2 (\tan (2 x) + \sec (2 x)) \sec (2 x) \tan (2 x)) - \sec (2 x) (\tan (2 x) + \sec (2 x)) (\sec (u_{4}) \tan (u_{4}) \frac{d}{d x} [2 x] + \frac{d}{d x} [\tan (2 x)])}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.11.3

Replace all occurrences of $u_{4}$ with $2 x$ .

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

Step 2.12

Differentiate.

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Step 2.12.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.12.2

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

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

Step 2.12.3

Simplify the expression.

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Step 2.12.3.1

Multiply $2$ by $1$ .

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

Step 2.12.3.2

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

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

Step 2.13

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

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Step 2.13.1

To apply the Chain Rule, set $u_{5}$ as $2 x$ .

$2 \frac{(\sec (2 x) + \tan (2 x)) (\sec (2 x) (2 \sec^{2} (2 x) + 2 \sec (2 x) \tan (2 x)) + 2 (\tan (2 x) + \sec (2 x)) \sec (2 x) \tan (2 x)) - \sec (2 x) (\tan (2 x) + \sec (2 x)) (2 \sec (2 x) \tan (2 x) + \frac{d}{d u_{5}} [\tan (u_{5})] \frac{d}{d x} [2 x])}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.13.2

The derivative of $\tan (u_{5})$ with respect to $u_{5}$ is $\sec^{2} (u_{5})$ .

$2 \frac{(\sec (2 x) + \tan (2 x)) (\sec (2 x) (2 \sec^{2} (2 x) + 2 \sec (2 x) \tan (2 x)) + 2 (\tan (2 x) + \sec (2 x)) \sec (2 x) \tan (2 x)) - \sec (2 x) (\tan (2 x) + \sec (2 x)) (2 \sec (2 x) \tan (2 x) + \sec^{2} (u_{5}) \frac{d}{d x} [2 x])}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.13.3

Replace all occurrences of $u_{5}$ with $2 x$ .

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

Step 2.14

Differentiate.

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Step 2.14.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.14.2

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

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

Step 2.14.3

Combine fractions.

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Step 2.14.3.1

Multiply $2$ by $1$ .

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

Step 2.14.3.2

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

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

Step 2.14.3.3

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

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

Step 2.15

Simplify.

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Step 2.15.1

Apply the distributive property.

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

Step 2.15.2

Apply the distributive property.

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

Step 2.15.3

Apply the distributive property.

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

Step 2.15.4

Apply the distributive property.

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

Step 2.15.5

Apply the distributive property.

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

Step 2.15.6

Apply the distributive property.

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

Step 2.15.7

Simplify the numerator.

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Step 2.15.7.1

Simplify each term.

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Step 2.15.7.1.1

Simplify each term.

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Step 2.15.7.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.1.2

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

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Step 2.15.7.1.1.2.1

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

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

Step 2.15.7.1.1.2.2

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

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Step 2.15.7.1.1.2.2.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.2.2.2

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

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

Step 2.15.7.1.1.2.3

Add $2$ and $1$ .

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

Step 2.15.7.1.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.1.4

Multiply $2 \sec (2 x) \sec (2 x)$ .

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Step 2.15.7.1.1.4.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.4.2

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.4.3

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

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

Step 2.15.7.1.1.4.4

Add $1$ and $1$ .

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

Step 2.15.7.1.1.5

Multiply $2 \tan (2 x) \sec (2 x) \tan (2 x)$ .

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Step 2.15.7.1.1.5.1

Raise $\tan (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.5.2

Raise $\tan (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.5.3

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

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

Step 2.15.7.1.1.5.4

Add $1$ and $1$ .

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

Step 2.15.7.1.1.6

Multiply $2 \sec (2 x) \sec (2 x)$ .

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Step 2.15.7.1.1.6.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.6.2

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.1.6.3

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

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

Step 2.15.7.1.1.6.4

Add $1$ and $1$ .

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

Step 2.15.7.1.2

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

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

Step 2.15.7.1.3

Expand $(\sec (2 x) + \tan (2 x)) (2 \sec^{3} (2 x) + 4 \sec^{2} (2 x) \tan (2 x) + 2 \tan^{2} (2 x) \sec (2 x))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2.15.7.1.4

Simplify each term.

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Step 2.15.7.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.4.2

Multiply $\sec (2 x)$ by $\sec^{3} (2 x)$ by adding the exponents.

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Step 2.15.7.1.4.2.1

Move $\sec^{3} (2 x)$ .

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

Step 2.15.7.1.4.2.2

Multiply $\sec^{3} (2 x)$ by $\sec (2 x)$ .

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Step 2.15.7.1.4.2.2.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.2.2.2

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

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

Step 2.15.7.1.4.2.3

Add $3$ and $1$ .

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

Step 2.15.7.1.4.3

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.4.4

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

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Step 2.15.7.1.4.4.1

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

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

Step 2.15.7.1.4.4.2

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

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Step 2.15.7.1.4.4.2.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.4.2.2

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

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

Step 2.15.7.1.4.4.3

Add $2$ and $1$ .

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

Step 2.15.7.1.4.5

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.4.6

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

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Step 2.15.7.1.4.6.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.6.2

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.6.3

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

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

Step 2.15.7.1.4.6.4

Add $1$ and $1$ .

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

Step 2.15.7.1.4.7

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.4.8

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.4.9

Multiply $4 \tan (2 x) \sec^{2} (2 x) \tan (2 x)$ .

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Step 2.15.7.1.4.9.1

Raise $\tan (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.9.2

Raise $\tan (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.9.3

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

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

Step 2.15.7.1.4.9.4

Add $1$ and $1$ .

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

Step 2.15.7.1.4.10

Rewrite using the commutative property of multiplication.

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

Step 2.15.7.1.4.11

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

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Step 2.15.7.1.4.11.1

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

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

Step 2.15.7.1.4.11.2

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

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Step 2.15.7.1.4.11.2.1

Raise $\tan (2 x)$ to the power of $1$ .

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

Step 2.15.7.1.4.11.2.2

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

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

Step 2.15.7.1.4.11.3

Add $2$ and $1$ .

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

Step 2.15.7.1.5

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

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

Step 2.15.7.1.6

Add $4 \sec^{3} (2 x) \tan (2 x)$ and $2 \sec^{3} (2 x) \tan (2 x)$ .

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

Step 2.15.7.1.7

Reorder the factors of $4 \tan^{2} (2 x) \sec^{2} (2 x)$ .

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

Step 2.15.7.1.8

Add $2 \sec^{2} (2 x) \tan^{2} (2 x)$ and $4 \sec^{2} (2 x) \tan^{2} (2 x)$ .

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

Step 2.15.7.1.9

Apply the distributive property.

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

Step 2.15.7.1.10

Simplify.

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Step 2.15.7.1.10.1

Multiply $2$ by $2$ .

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

Step 2.15.7.1.10.2

Multiply $6$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 (\sec^{3} (2 x) \tan (2 x)) + 2 (6 \sec^{2} (2 x) \tan^{2} (2 x)) + 2 (2 \tan^{3} (2 x) \sec (2 x)) + 2 ((- \sec (2 x) \tan (2 x) - \sec (2 x) \sec (2 x)) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.10.3

Multiply $6$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 (\sec^{3} (2 x) \tan (2 x)) + 12 (\sec^{2} (2 x) \tan^{2} (2 x)) + 2 (2 \tan^{3} (2 x) \sec (2 x)) + 2 ((- \sec (2 x) \tan (2 x) - \sec (2 x) \sec (2 x)) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.10.4

Multiply $2$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 (\sec^{3} (2 x) \tan (2 x)) + 12 (\sec^{2} (2 x) \tan^{2} (2 x)) + 4 (\tan^{3} (2 x) \sec (2 x)) + 2 ((- \sec (2 x) \tan (2 x) - \sec (2 x) \sec (2 x)) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.11

Remove parentheses.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 ((- \sec (2 x) \tan (2 x) - \sec (2 x) \sec (2 x)) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.12

Multiply $- \sec (2 x) \sec (2 x)$ .

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Step 2.15.7.1.12.1

Raise $\sec (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 ((- \sec (2 x) \tan (2 x) - (\sec^{1} (2 x) \sec (2 x))) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.12.2

Raise $\sec (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 ((- \sec (2 x) \tan (2 x) - (\sec^{1} (2 x) \sec^{1} (2 x))) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.12.3

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 ((- \sec (2 x) \tan (2 x) - {\sec (2 x)}^{1 + 1}) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.12.4

Add $1$ and $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 ((- \sec (2 x) \tan (2 x) - \sec^{2} (2 x)) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.13

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

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Step 2.15.7.1.13.1

Apply the distributive property.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- \sec (2 x) \tan (2 x) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.13.2

Apply the distributive property.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- \sec (2 x) \tan (2 x) (2 \sec (2 x) \tan (2 x)) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x) + 2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.13.3

Apply the distributive property.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- \sec (2 x) \tan (2 x) (2 \sec (2 x) \tan (2 x)) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14

Simplify and combine like terms.

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Step 2.15.7.1.14.1

Simplify each term.

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Step 2.15.7.1.14.1.1

Multiply $- \sec (2 x) \tan (2 x) (2 \sec (2 x) \tan (2 x))$ .

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Step 2.15.7.1.14.1.1.1

Multiply $2$ by $- 1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec (2 x) \tan (2 x) (\sec (2 x) \tan (2 x)) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.2

Raise $\sec (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 (\sec^{1} (2 x) \sec (2 x)) \tan (2 x) \tan (2 x) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.3

Raise $\sec (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 (\sec^{1} (2 x) \sec^{1} (2 x)) \tan (2 x) \tan (2 x) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.4

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 {\sec (2 x)}^{1 + 1} \tan (2 x) \tan (2 x) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.5

Add $1$ and $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan (2 x) \tan (2 x) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.6

Raise $\tan (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) (\tan^{1} (2 x) \tan (2 x)) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.7

Raise $\tan (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) (\tan^{1} (2 x) \tan^{1} (2 x)) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.8

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) {\tan (2 x)}^{1 + 1} - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.1.9

Add $1$ and $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - \sec (2 x) \tan (2 x) (2 \sec^{2} (2 x)) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.2

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

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Step 2.15.7.1.14.1.2.1

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - (\sec^{2} (2 x) \sec (2 x)) \tan (2 x) \cdot 2 - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.2.2

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

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Step 2.15.7.1.14.1.2.2.1

Raise $\sec (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - (\sec^{2} (2 x) \sec^{1} (2 x)) \tan (2 x) \cdot 2 - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.2.2.2

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - {\sec (2 x)}^{2 + 1} \tan (2 x) \cdot 2 - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.2.3

Add $2$ and $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - \sec^{3} (2 x) \tan (2 x) \cdot 2 - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.3

Multiply $2$ by $- 1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - \sec^{2} (2 x) (2 \sec (2 x) \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.4

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

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Step 2.15.7.1.14.1.4.1

Move $\sec (2 x)$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - (\sec (2 x) \sec^{2} (2 x)) (2 \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.4.2

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

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Step 2.15.7.1.14.1.4.2.1

Raise $\sec (2 x)$ to the power of $1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - (\sec^{1} (2 x) \sec^{2} (2 x)) (2 \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.4.2.2

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - {\sec (2 x)}^{1 + 2} (2 \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.4.3

Add $1$ and $2$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - \sec^{3} (2 x) (2 \tan (2 x)) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.5

Multiply $2$ by $- 1$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - \sec^{2} (2 x) (2 \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.6

Rewrite using the commutative property of multiplication.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 1 \cdot 2 \sec^{2} (2 x) \sec^{2} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.7

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

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Step 2.15.7.1.14.1.7.1

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 1 \cdot 2 (\sec^{2} (2 x) \sec^{2} (2 x)))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.7.2

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 1 \cdot 2 {\sec (2 x)}^{2 + 2})}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.7.3

Add $2$ and $2$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 1 \cdot 2 \sec^{4} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.1.8

Multiply $- 1$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{4} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.14.2

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

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x) - 4 \sec^{3} (2 x) \tan (2 x) - 2 \sec^{4} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.15

Apply the distributive property.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) + 2 (- 2 \sec^{2} (2 x) \tan^{2} (2 x)) + 2 (- 4 \sec^{3} (2 x) \tan (2 x)) + 2 (- 2 \sec^{4} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.16

Simplify.

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Step 2.15.7.1.16.1

Multiply $- 2$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 (\sec^{2} (2 x) \tan^{2} (2 x)) + 2 (- 4 \sec^{3} (2 x) \tan (2 x)) + 2 (- 2 \sec^{4} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.16.2

Multiply $- 4$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 (\sec^{2} (2 x) \tan^{2} (2 x)) - 8 (\sec^{3} (2 x) \tan (2 x)) + 2 (- 2 \sec^{4} (2 x))}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.16.3

Multiply $- 2$ by $2$ .

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 (\sec^{2} (2 x) \tan^{2} (2 x)) - 8 (\sec^{3} (2 x) \tan (2 x)) - 4 \sec^{4} (2 x)}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.1.17

Remove parentheses.

$\frac{4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 \sec^{2} (2 x) \tan^{2} (2 x) - 8 \sec^{3} (2 x) \tan (2 x) - 4 \sec^{4} (2 x)}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.2

Combine the opposite terms in $4 \sec^{4} (2 x) + 12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 \sec^{2} (2 x) \tan^{2} (2 x) - 8 \sec^{3} (2 x) \tan (2 x) - 4 \sec^{4} (2 x)$ .

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Step 2.15.7.2.1

Subtract $4 \sec^{4} (2 x)$ from $4 \sec^{4} (2 x)$ .

$\frac{12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 \sec^{2} (2 x) \tan^{2} (2 x) - 8 \sec^{3} (2 x) \tan (2 x) + 0}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.2.2

Add $12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 \sec^{2} (2 x) \tan^{2} (2 x) - 8 \sec^{3} (2 x) \tan (2 x)$ and $0$ .

$\frac{12 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 \sec^{2} (2 x) \tan^{2} (2 x) - 8 \sec^{3} (2 x) \tan (2 x)}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.3

Subtract $8 \sec^{3} (2 x) \tan (2 x)$ from $12 \sec^{3} (2 x) \tan (2 x)$ .

$\frac{4 \sec^{3} (2 x) \tan (2 x) + 12 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x) - 4 \sec^{2} (2 x) \tan^{2} (2 x)}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.7.4

Subtract $4 \sec^{2} (2 x) \tan^{2} (2 x)$ from $12 \sec^{2} (2 x) \tan^{2} (2 x)$ .

$\frac{4 \sec^{3} (2 x) \tan (2 x) + 8 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x)}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.8

Simplify the numerator.

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Step 2.15.8.1

Factor $4 \sec (2 x) \tan (2 x)$ out of $4 \sec^{3} (2 x) \tan (2 x) + 8 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x)$ .

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Step 2.15.8.1.1

Factor $4 \sec (2 x) \tan (2 x)$ out of $4 \sec^{3} (2 x) \tan (2 x)$ .

$\frac{4 \sec (2 x) \tan (2 x) (\sec^{2} (2 x)) + 8 \sec^{2} (2 x) \tan^{2} (2 x) + 4 \tan^{3} (2 x) \sec (2 x)}{{(\sec (2 x) + \tan (2 x))}^{2}}$

Step 2.15.8.1.2

Factor $4 \sec (2 x) \tan (2 x)$ out of $8 \sec^{2} (2 x) \tan^{2} (2 x)$ .

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

Step 2.15.8.1.3

Factor $4 \sec (2 x) \tan (2 x)$ out of $4 \tan^{3} (2 x) \sec (2 x)$ .

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

Step 2.15.8.1.4

Factor $4 \sec (2 x) \tan (2 x)$ out of $4 \sec (2 x) \tan (2 x) (\sec^{2} (2 x)) + 4 \sec (2 x) \tan (2 x) (2 \sec (2 x) \tan (2 x))$ .

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

Step 2.15.8.1.5

Factor $4 \sec (2 x) \tan (2 x)$ out of $4 \sec (2 x) \tan (2 x) (\sec^{2} (2 x) + 2 \sec (2 x) \tan (2 x)) + 4 \sec (2 x) \tan (2 x) (\tan^{2} (2 x))$ .

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

Step 2.15.8.2

Factor using the perfect square rule.

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Step 2.15.8.2.1

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

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

Step 2.15.8.2.2

Rewrite the polynomial.

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

Step 2.15.8.2.3

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

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

Step 2.15.9

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

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Step 2.15.9.1

Cancel the common factor.

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

Step 2.15.9.2

Divide $4 \sec (2 x) \tan (2 x)$ by $1$ .

$f'' (x) = 4 \sec (2 x) \tan (2 x)$