Answer
\[ = 5{\sin ^4}x\cos x\]
Work Step by Step
\[\begin{gathered}
y = {\sin ^5}x \hfill \\
\hfill \\
y = f\,\left( u \right) = {u^5} \hfill \\
\hfill \\
set\,\,\,u = g\,\left( x \right) = \sin x \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,1\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
{\text{ }}\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}} \hfill \\
\hfill \\
Therefore \hfill \\
\hfill \\
\,\,\,\frac{{dy}}{{dx}} = \frac{d}{{du}}\,\left( {{u^5}} \right) \cdot \frac{d}{{dx}}\,\left( {\sin x} \right) \hfill \\
\hfill \\
= 5{u^4}\cos x \hfill \\
\hfill \\
substitute\,\,\,back\,\,u \hfill \\
\hfill \\
= 5{\sin ^4}x\cos x \hfill \\
\end{gathered} \]
