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