Answer
\[{f^,}\,\left( x \right) = \frac{3}{2}{x^{\frac{1}{2}}}\,\left( {2x - 3} \right) + 2{x^{\frac{3}{2}}}\]
Work Step by Step
\[\begin{gathered}
f\,\left( x \right) = \,\left( {2x - 3} \right){x^{\frac{3}{2}}} \hfill \\
\hfill \\
Use\,\,the\,\,product\,\,rule \hfill \\
\hfill \\
{f^,}\,\left( x \right) = \,\left( {2x - 3} \right)\,{\left( {{x^{\frac{3}{2}}}} \right)^,} + {x^{\frac{3}{2}}}\,{\left( {2x - 3} \right)^,} \hfill \\
\hfill \\
differentiate \hfill \\
\hfill \\
{f^,}\,\left( x \right) = \,\left( {2x - 3} \right)\,\left( {\frac{3}{2}{x^{\frac{1}{2}}}} \right) + {x^{\frac{3}{2}}}\,\left( 2 \right) \hfill \\
\hfill \\
multiply \hfill \\
\hfill \\
{f^,}\,\left( x \right) = \frac{3}{2}{x^{\frac{1}{2}}}\,\left( {2x - 3} \right) + 2{x^{\frac{3}{2}}} \hfill \\
\hfill \\
\end{gathered} \]
