Derivative of Cosecant Function

Theorem

 * $D_x \left({\csc x}\right) = -\csc x \cot x$, when $\sin x \ne 0$.

Proof

 * From the definition of the cosecant function, $\csc x = \dfrac 1 {\sin x}$.
 * From Derivative of Sine Function we have $D_x \left({\sin x}\right) = \cos x$.

Then:

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