Answer
\[{y^,} = \frac{{5{x^4}}}{{\,{{\left( {x + 1} \right)}^6}}}\]
Work Step by Step
\[\begin{gathered}
y = \,{\left( {\frac{x}{{x + 1}}} \right)^5} \hfill \\
\hfill \\
Differentiate\,\,both\,\,sides \hfill \\
\hfill \\
{y^,} = \frac{d}{{dx}}\,{\left( {\frac{x}{{x + 1}}} \right)^5} \hfill \\
\hfill \\
use\,\,the\,\,chain\,\,rule\,\,for\,\,powers \hfill \\
\hfill \\
\frac{d}{{dx}}\left( {g{{\left( x \right)}^n}} \right) = ng{\left( x \right)^{n - 1}}g'\left( x \right) \hfill \\
\hfill \\
Therefore, \hfill \\
\hfill \\
{y^,} = 5\,{\left( {\frac{x}{{x + 1}}} \right)^4} \cdot \,{\left( {\frac{x}{{x + 1}}} \right)^,} \hfill \\
\hfill \\
use\,\,the\,\,quotient\,\,rule \hfill \\
\hfill \\
{y^,} = 5\,{\left( {\frac{x}{{x + 1}}} \right)^4} \cdot \frac{{x + 1 - x}}{{\,{{\left( {x + 1} \right)}^2}}} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
{y^,} = \frac{{5{x^4}}}{{\,{{\left( {x + 1} \right)}^6}}} \hfill \\
\end{gathered} \]
