Derivative of Cotangent Function

Theorem

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

where $\sin x \ne 0$.

Proof
From the definition of the cotangent function:
 * $\cot x = \dfrac {\cos x} {\sin x}$

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

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

Then:

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

The result follows from the definition of the cosecant function.