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:
 * $\csc x = \dfrac 1 {\sin x}$

From Derivative of Sine Function:
 * $D_x \left({\sin x}\right) = \cos x$

Then:

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