Answer
$-\csc^2 x$
Work Step by Step
Let us suppose that $z(x)= \cot x$
Use reciprocal identity $\cot x=\dfrac{\cos x}{\sin x}$
We differentiate both sides with respect to $x$.
$z^{\prime}(x)=\dfrac{d}{dx} [ \dfrac{\cos x}{\sin x}] \\=\dfrac{\sin x (-\sin x) -\cos x (\cos x) }{(\sin x)^2}$
Simplify to obtain:
$z^{\prime}(x)=-\dfrac{\sin^2 x+\cos^2 x}{(\sin x)^2} =-\csc^2 x$
