# Derivative of Cosecant Function

## Theorem

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

where $\sin x \ne 0$.

## Proof

From the definition of the cosecant function:

$\csc x = \dfrac 1 {\sin x}$
$D_x \left({\sin x}\right) = \cos x$

Then:

 $\displaystyle D_x \left({\csc x}\right)$ $=$ $\displaystyle \cos x \frac {-1} {\sin^2 x}$ Chain Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \frac {-1} {\sin x} \frac {\cos x} {\sin x}$ $\displaystyle$ $=$ $\displaystyle -\csc x \cot x$ Definitions of secant and cotangent

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

$\blacksquare$