Answer
\[{f^,}\,\left( s \right) = - \frac{{{e^s}}}{{1 + {e^{2s}}}}\]
Work Step by Step
\[\begin{gathered}
f\,\left( s \right) = {\cot ^{ - 1}}\,\left( {{e^s}} \right) \hfill \\
\hfill \\
Use\,\,\frac{d}{{ds}}\,\,\left[ {{{\cot }^{ - 1}}u} \right] = \frac{{ - {u^,}}}{{1 + {u^2}}} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
{f^,}\,\left( s \right) = \frac{{ - {e^s}}}{{1 + \,{{\left( {{e^s}} \right)}^2}}} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
{f^,}\,\left( s \right) = - \frac{{{e^s}}}{{1 + {e^{2s}}}} \hfill \\
\end{gathered} \]
