Calculus Examples

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

Step 1

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

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

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 $g (x) = \sec (x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 5

Multiply $\cos (x)$ by $1$ .

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

Step 6

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

$2 \cos (x) (\sec (x) \tan (x))$

Step 7

Simplify.

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Step 7.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 7.1.1

Add parentheses.

$2 (\cos (x) \sec (x)) \tan (x)$

Step 7.1.2

Reorder $\cos (x)$ and $\sec (x)$ .

$2 (\sec (x) \cos (x)) \tan (x)$

Step 7.1.3

Rewrite $2 \cos (x) \sec (x)$ in terms of sines and cosines.

$2 (\frac{1}{\cos (x)} \cos (x)) \tan (x)$

Step 7.1.4

Cancel the common factors.

$2 \cdot 1 \tan (x)$

Step 7.2

Multiply $2$ by $1$ .

$2 \tan (x)$

Step 7.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$2 \frac{\sin (x)}{\cos (x)}$

Step 7.4

Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{2 \sin (x)}{\cos (x)}$

Step 7.5

Separate fractions.

$\frac{2}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 7.6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{2}{1} \tan (x)$

Step 7.7

Divide $2$ by $1$ .

$2 \tan (x)$