Answer
\[{p^,} = \frac{{1250{e^{ - 0.5t}}}}{{\,{{\left( {12 + 5{e^{ - 0.5t}}} \right)}^2}}}\]
Work Step by Step
\[\begin{gathered}
p = \frac{{500}}{{12 + 5{e^{ - 0.5t}}}} \hfill \\
Find\,\,the\,\,derivative \hfill \\
{p^,} = \frac{d}{{dt}}\,\,\left[ {\,\frac{{500}}{{12 + 5{e^{ - 0.5t}}}}} \right] \hfill \\
Use\,\,the\,\,quotient\,\,rule \hfill \\
{p^,} = \frac{{\,\left( {12 + 5{e^{ - 0.5t}}} \right)\,{{\left( {500} \right)}^,} - 500\,{{\left( {12 + 5{e^{ - 0.5t}}} \right)}^,}}}{{\,{{\left( {12 + 5{e^{ - 0.5t}}} \right)}^2}}} \hfill \\
Then \hfill \\
{p^,} = \frac{{\,\left( {12 + 5{e^{ - 0.5t}}} \right)\,\left( 0 \right) - 500\,\left( {5{e^{ - 0.5t}}} \right)\,\left( { - 0.5} \right)}}{{\,{{\left( {12 + 5{e^{ - 0.5t}}} \right)}^2}}} \hfill \\
Multiply \hfill \\
{p^,} = \frac{{1250{e^{ - 0.5t}}}}{{\,{{\left( {12 + 5{e^{ - 0.5t}}} \right)}^2}}} \hfill \\
\end{gathered} \]
