Calculus Examples

$\ln (\cot (x))$

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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