Calculus Examples

$y = \ln (\tan (x))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (\tan (x)))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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 3.4

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

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

Step 3.5

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

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

Step 3.6

Simplify.

Tap for more steps...

Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

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

Tap for more steps...

Step 3.6.5.1

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

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

Step 3.6.5.2

Cancel the common factor.

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

Step 3.6.5.3

Rewrite the expression.

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

Step 3.6.6

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

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

Step 3.6.7

Separate fractions.

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

Step 3.6.8

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

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

Step 3.6.9

Separate fractions.

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

Step 3.6.10

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

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

Step 3.6.11

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

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

Step 3.6.12

One to any power is one.

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

Step 3.6.13

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \csc (x) \sec (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \csc (x) \sec (x)$