### Derivative of Cotangent

Let $y = f(x) = \cot x$.

Since $\cot x = {{\cos x} \over {\sin x}}$, we use the quotient rule to get

${{dy} \over {dx}} = {{({d \over {dx}}(\cos x))(\sin x) - (\cos x)({d \over {dx}}(\sin x))} \over {(\sin x)^2 }}$

$= {{( - \sin x)(\sin x) - (\cos x)(\cos x)} \over {(\sin x)^2 }}$

$= {{ - \sin ^2 x - \cos ^2 x} \over {\sin ^2 x}}$

$= {{ - 1} \over {\sin ^2 x}}$

$= - \csc ^2 x$