Calculus Examples

$\ln (\tan (x))$

Step 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 $g (x) = \tan (x)$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

$\frac{1}{\frac{\sin (x)}{\cos (x)}} \frac{d}{d x} [\tan (x)]$

Step 3

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

$\frac{\cos (x)}{\sin (x)} \frac{d}{d x} [\tan (x)]$

Step 4

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

$\cot (x) \frac{d}{d x} [\tan (x)]$

Step 5

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

$\cot (x) \sec^{2} (x)$

Step 6

Simplify.

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Step 6.1

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

$\sec^{2} (x) \cot (x)$

Step 6.2

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

${(\frac{1}{\cos (x)})}^{2} \cot (x)$

Step 6.3

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

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

Step 6.4

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 6.5

Cancel the common factor of $\cos (x)$ .

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Step 6.5.1

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

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

Step 6.5.2

Cancel the common factor.

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

Step 6.5.3

Rewrite the expression.

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

Step 6.6

Multiply $\frac{1^{2}}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 6.7

Separate fractions.

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

Step 6.8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 6.9

Separate fractions.

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

Step 6.10

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 6.11

Divide $1^{2}$ by $1$ .

$1^{2} \csc (x) \sec (x)$

Step 6.12

One to any power is one.

$1 \csc (x) \sec (x)$

Step 6.13

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

$\csc (x) \sec (x)$