#### Answer

\[\frac{{dy}}{{dx}} = x{\cos ^2}x - x{\sin ^2}x + \cos x\sin x\]

#### Work Step by Step

\[\begin{gathered}
y = x\cos x\sin x \hfill \\
\hfill \\
by\,\,the\,\,product\,\,rule \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = x\,\left( {\cos x\cos x + \sin x\,\left( { - \sin x} \right)} \right) + \cos x\sin x \hfill \\
\hfill \\
{\text{Therefore}}{\text{,}} \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = x\,\left( {{{\cos }^2}x - {{\sin }^2}x} \right) + \cos x\sin x \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = x{\cos ^2}x - x{\sin ^2}x + \cos x\sin x \hfill \\
\end{gathered} \]