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

Then:

$$ $$ $$

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