Derivative of Cotangent Function

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

Proof

 * From the definition of the cotangent function, $$\cot x = \frac {\cos x} {\sin x}$$.
 * 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.