#### Answer

\[{y^,} = \frac{5}{{2\ln 5\,\left( {5x + 2} \right)}}\]

#### Work Step by Step

\[\begin{gathered}
y = {\log _5}\sqrt {5x + 2} \hfill \\
Write\,\,\sqrt {5x + 2} \,\,as\,\,\,{\left( {5x + 2} \right)^{1/2}} \hfill \\
y = {\log _5}\,{\left( {5x + 2} \right)^{1/2}} \hfill \\
Use\,\,\log \,\,property \hfill \\
y = \frac{1}{2}{\log _5}\,\left( {5x + 2} \right) \hfill \\
Find\,\,the\,\,derivative \hfill \\
{y^,} = \frac{1}{2}\,\,{\left[ {{{\log }_5}\,\left( {5x + 2} \right)} \right]^,} \hfill \\
Use\,\,the\,\,formula \hfill \\
\frac{d}{{dx}}\,\,\left[ {{{\log }_a}\left| {g\,\left( x \right)} \right|} \right] = \frac{1}{{\ln a}} \cdot \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\
Then \hfill \\
{y^,} = \frac{1}{{2\ln 5}}\,\left( {\frac{5}{{5x + 2}}} \right) \hfill \\
{y^,} = \frac{5}{{2\ln 5\,\left( {5x + 2} \right)}} \hfill \\
\end{gathered} \]