Answer
\[{p^,} = \frac{{8,000{e^{ - 0.2t}}}}{{\,{{\left( {9 + 4{e^{ - 0.2t}}} \right)}^2}}}\]
Work Step by Step
\[\begin{gathered}
p = \frac{{10,000}}{{9 + 4{e^{ - 0.2t}}}} \hfill \\
Find\,\,the\,\,derivative \hfill \\
{p^,} = \frac{d}{{dt}}\,\,\left[ {\frac{{10,000}}{{9 + 4{e^{ - 0.2t}}}}} \right] \hfill \\
Use\,\,the\,\,quotient\,\,rule \hfill \\
{p^,} = \frac{{\,\left( {9 + 4{e^{ - 0.2t}}} \right)\,{{\left( {10,000} \right)}^,} - 10,000\,{{\left( {9 + 4{e^{ - 0.2t}}} \right)}^,}}}{{\,{{\left( {9 + 4{e^{ - 0.2t}}} \right)}^2}}} \hfill \\
Then \hfill \\
{p^,} = \frac{{\,\left( {9 + 4{e^{ - 0.2t}}} \right)\,\left( 0 \right) - 10,000\,\left( {4{e^{ - 0.2t}}} \right)\,\left( { - 0.2} \right)}}{{\,{{\left( {9 + 4{e^{ - 0.2t}}} \right)}^2}}} \hfill \\
Multiply \hfill \\
{p^,} = \frac{{8,000{e^{ - 0.2t}}}}{{\,{{\left( {9 + 4{e^{ - 0.2t}}} \right)}^2}}} \hfill \\
\end{gathered} \]
