Answer
$-\csc x\cot x$
Work Step by Step
Let us suppose that $z(x)= \csc x$
Use reciprocal identity $\csc x=\dfrac{1}{\sin x}$
We differentiate both sides with respect to $x$.
$z^{\prime}(x)=\dfrac{d}{dx} [\dfrac{1}{\sin x}] \\=-(\sin x)^{-1-1} \dfrac{d}{dx} [\sin x]$
Simplify to obtain:
$z^{\prime}(x)=-(\sin x)^{-2}\times \cos x=-\csc x\cot x$
