Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.10 Derivatives of Inverse Trigonometric Functions - 3.10 Execises: 29

Answer

\[{f^,}\,\left( s \right) = - \frac{{{e^s}}}{{1 + {e^{2s}}}}\]

Work Step by Step

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