Answer
\[{y^,} = 2{e^{2x \cdot \,}} \cdot \,\,\,{\left( {2x - 7} \right)^5} + 10{e^{2x}}\, \cdot \,\,{\left( {2x - 7} \right)^4}\]
Work Step by Step
\[\begin{gathered}
y = {e^{2x}}\,{\left( {2x - 7} \right)^5} \hfill \\
\hfill \\
Product\,\,rule \hfill \\
\hfill \\
{y^,} = \,{\left( {{e^{2x}}} \right)^,} \cdot \,{\left( {2x - 7} \right)^5} + {e^{2x}} \cdot \,{\left( {\,{{\left( {2x - 7} \right)}^5}} \right)^,} \hfill \\
\hfill \\
Chain\,\,rule \hfill \\
\hfill \\
{y^,} = {e^{2x}}\,{\left( {2x} \right)^,} \cdot \,\,{\left( {2x - 7} \right)^5} + {e^{2x}} \cdot 5\,{\left( {2x - 7} \right)^4} \cdot \,\,{\left( {2x - 7} \right)^,} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
{y^,} = 2{e^{2x}} \cdot \,{\left( {2x - 7} \right)^5} + 5{e^{2x}} \cdot \,{\left( {2x - 7} \right)^4} \cdot \,\,\,2 \hfill \\
\hfill \\
multiply \hfill \\
\hfill \\
{y^,} = 2{e^{2x \cdot \,}} \cdot \,\,\,{\left( {2x - 7} \right)^5} + 10{e^{2x}}\, \cdot \,\,{\left( {2x - 7} \right)^4} \hfill \\
\end{gathered} \]
