# Derivative of Cotangent Function

## Theorem

$D_x \left({\cot x}\right) = -\csc^2 x = \dfrac {-1} {\sin^2 x}$

where $\sin x \ne 0$.

### Corollary

$D_x \left({\cot a x}\right) = -a \csc^2 a x$

## Proof

From the definition of the cotangent function:

$\cot x = \dfrac {\cos x} {\sin x}$
$D_x \left({\sin x}\right) = \cos x$
$D_x \left({\cos x}\right) = -\sin x$

Then:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle D_x \left({\cot x}\right)$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \frac {\sin x \left({-\sin x}\right) - \cos x \cos x} {\sin^2 x}$$ $$\displaystyle$$ $$\displaystyle$$ Quotient Rule for Derivatives $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \frac {-\left({\sin^2 x + \cos^2 x}\right)} {\sin^2 x}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \frac {-1} {\sin^2 x}$$ $$\displaystyle$$ $$\displaystyle$$ Sum of Squares of Sine and Cosine

This is valid only when $\sin x \ne 0$.

The result follows from the definition of the cosecant function.

$\blacksquare$