Answer
\[{y^,} = \frac{1}{4} \cdot \,{\left( {\frac{{2x}}{{4x - 3}}} \right)^{ - \frac{3}{4}}} \cdot \frac{{ - 6}}{{\,{{\left( {4x - 3} \right)}^2}}}\]
Work Step by Step
\[\begin{gathered}
y = \sqrt[4]{{\frac{{2x}}{{4x - 3}}}} \hfill \\
\hfill \\
derivative \hfill \\
\hfill \\
{y^,} = \,{\left( {\frac{{2x}}{{4x - 3}}} \right)^{\frac{1}{4}}} \hfill \\
\hfill \\
use\,\,the\,\,chain\,rule \hfill \\
\hfill \\
{y^,} = \frac{1}{4} \cdot \,{\left( {\frac{{2x}}{{4x - 3}}} \right)^{ - \frac{3}{4}}} \cdot \,{\left( {\frac{{2x}}{{4x - 3}}} \right)^,} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
{y^,} = \frac{1}{4} \cdot \,{\left( {\frac{{2x}}{{4x - 3}}} \right)^{ - \frac{3}{4}}} \cdot \,\frac{{2\,\left( {4x - 3} \right) - 2x \cdot 4}}{{\,{{\left( {4x - 3} \right)}^2}}} \hfill \\
\hfill \\
{y^,} = \frac{1}{4} \cdot \,{\left( {\frac{{2x}}{{4x - 3}}} \right)^{ - \frac{3}{4}}} \cdot \frac{{ - 6}}{{\,{{\left( {4x - 3} \right)}^2}}} \hfill \\
\end{gathered} \]
