Derivative of Cotangent Function

Theorem

 * $\displaystyle D_x \left({\cot x}\right) = -\csc^2 x = \frac {-1} {\sin^2 x}$, when $\sin x \ne 0$.

Proof

 * From the definition of the cotangent function, $\cot x = \dfrac {\cos x} {\sin x}$.
 * From Derivative of Sine Function we have $D_x \left({\sin x}\right) = \cos x$.
 * From Derivative of Cosine Function we have $D_x \left({\cos x}\right) = -\sin x$.

Then:

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

The result follows from the definition of the cosecant function.