Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate using the Sum Rule.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Combine the numerators over the common denominator.

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

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

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

Step 6.7.3

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

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