Answer
\[ = \frac{{30\,{{\left( {3x} \right)}^4}}}{{\,{{\left( {4x + 2} \right)}^6}}}\]
Work Step by Step
\[\begin{gathered}
y = \,{\left( {\frac{{3x}}{{4x + 2}}} \right)^5} \hfill \\
\hfill \\
differentiate \hfill \\
\hfill \\
{y^,} = \,{\left( {\frac{{3x}}{{4x + 2}}} \right)^5} \hfill \\
\hfill \\
use\,\,the\,Chain\,\,rule \hfill \\
\hfill \\
= 5\,{\left( {\frac{{3x}}{{4x + 2}}} \right)^4} \cdot \,\,{\left( {\frac{{3x}}{{4x + 2}}} \right)^,} \hfill \\
\hfill \\
Quotient\,\,rule \hfill \\
\hfill \\
= 5\,{\left( {\frac{{3x}}{{4x + 2}}} \right)^4} \cdot \,\frac{{3\,\left( {4x + 2} \right) - 4\,\left( {3x} \right)}}{{\,{{\left( {4x + 2} \right)}^2}}} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= 5\,{\left( {\frac{{3x}}{{4x + 2}}} \right)^4} \cdot \,\,\frac{6}{{\,{{\left( {4x + 2} \right)}^2}}} \hfill \\
\hfill \\
= \frac{{30\,{{\left( {3x} \right)}^4}}}{{\,{{\left( {4x + 2} \right)}^6}}} \hfill \\
\end{gathered} \]
