Answer
\[{y^,} = \frac{1}{{2\,\left( {x + 5} \right)}}\]
Work Step by Step
\[\begin{gathered}
y = \ln \sqrt {x + 5} \hfill \\
Write\,\,\sqrt {x + 5} \,\,as\,\,\,{\left( {x + 5} \right)^{1/2}} \hfill \\
y = \ln \,\,\,{\left( {x + 5} \right)^{1/2}} \hfill \\
Use\,\,the\,\,\log \,\,property \hfill \\
\ln {a^n} = n\ln a \hfill \\
y = \frac{1}{2}\ln \,\left( {x + 5} \right) \hfill \\
Differentiating \hfill \\
{y^,} = \,\,{\left[ {\frac{1}{2}\ln \,\left( {x + 5} \right)} \right]^,} \hfill \\
{y^,} = \frac{1}{2}\,\,{\left[ {\ln \,\left( {x + 5} \right)} \right]^,} \hfill \\
Use\,\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\
{y^,} = \frac{1}{2}\,\left( {\frac{1}{{x + 5}}} \right) \hfill \\
{y^,} = \frac{1}{{2\,\left( {x + 5} \right)}} \hfill \\
\hfill \\
\end{gathered} \]
