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