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: 26

Answer

\[{f^,}\,\left( x \right) = \frac{{4{e^{4x}}}}{{1 + {e^{8x}}}}\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = {\tan ^{ - 1}}\,\left( {{e^{4x}}} \right) \hfill \\ \hfill \\ Use\,\,\frac{d}{{dy}}\,\,\,\left[ {{{\tan }^{ - 1}}u} \right] = \frac{{{u^,}}}{{1 + {u^2}}} \hfill \\ then \hfill \\ \hfill \\ {f^,}\,\left( x \right) = \frac{{\,\left( {{e^{4x}}} \right)}}{{1 + \,{{\left( {{e^{4x}}} \right)}^2}}} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ {f^,}\,\left( x \right) = \frac{{4{e^{4x}}}}{{1 + {e^{8x}}}} \hfill \\ \end{gathered} \]
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