Answer
$$\frac{{dy}}{{dx}} = x\csc x\left( {2 - x\cot x} \right)$$
Work Step by Step
$$\eqalign{
& y = {x^2}\csc x \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = {D_x}\left( {{x^2}\csc x} \right) \cr
& {\text{use the product rule}} \cr
& \frac{{dy}}{{dx}} = {x^2} \cdot {D_x}\left( {\csc x} \right) + \csc x \cdot {D_x}\left( {{x^2}} \right) \cr
& {\text{solve derivatives }} \cr
& \frac{{dy}}{{dx}} = {x^2}\left( { - \csc x\cot x} \right) + \csc x\left( {2x} \right) \cr
& {\text{multiply}} \cr
& \frac{{dy}}{{dx}} = - {x^2}\csc x\cot x + 2x\csc x \cr
& {\text{factoring}} \cr
& \frac{{dy}}{{dx}} = x\csc x\left( {2 - x\cot x} \right) \cr} $$
