Calculus Examples

$f (x) = \ln (\tan (x) \cdot \tan (x))$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Add $1$ and $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Simplify with factoring out.

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Step 6.1

Rewrite $1$ as $1^{2}$ .

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

Step 6.2

Rewrite $\frac{1^{2}}{\tan^{2} (x)}$ as ${(\frac{1}{\tan (x)})}^{2}$ .

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

Step 7

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

${(\frac{1}{\frac{\sin (x)}{\cos (x)}})}^{2} \frac{d}{d x} [\tan^{2} (x)]$

Step 8

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

${(\frac{\cos (x)}{\sin (x)})}^{2} \frac{d}{d x} [\tan^{2} (x)]$

Step 9

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

$\cot^{2} (x) \frac{d}{d x} [\tan^{2} (x)]$

Step 10

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

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Step 10.1

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

$\cot^{2} (x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\tan (x)])$

Step 10.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\cot^{2} (x) (2 u_{2} \frac{d}{d x} [\tan (x)])$

Step 10.3

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

$\cot^{2} (x) (2 \tan (x) \frac{d}{d x} [\tan (x)])$

Step 11

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

$2 \cdot \cot^{2} (x) \tan (x) \frac{d}{d x} [\tan (x)]$

Step 12

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

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

Step 13

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

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