Calculus Examples

$f (x) = \frac{1}{\tan^{2} (x)}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$2 \cot (x) (- \csc^{2} (x))$

Step 8

Simplify the expression.

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Step 8.1

Multiply $- 1$ by $2$ .

$- 2 \cot (x) \csc^{2} (x)$

Step 8.2

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

$- 2 \csc^{2} (x) \cot (x)$