#### Answer

\[\frac{{dy}}{{dx}} = \,\left( {300{x^5} - 225{x^2}} \right)\,{\left( {2{x^6} - 3{x^3} + 3} \right)^{24}}\]

#### Work Step by Step

\[\begin{gathered}
y = \,{\left( {2{x^6} - 3{x^3} + 3} \right)^{25}} \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,1\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}} \hfill \\
\hfill \\
set\,\,u = 2{x^6} - 3{x^3} + 3 \hfill \\
\hfill \\
\frac{{du}}{{dx}} = 12{x^5} - 9{x^2} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = 25\,{\left( {2{x^6} - 3{x^3} + 3} \right)^{24}}\,\left( {12{x^5} - 9{x^2}} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \,\left( {300{x^5} - 225{x^2}} \right)\,{\left( {2{x^6} - 3{x^3} + 3} \right)^{24}} \hfill \\
\end{gathered} \]