Calculus: Early Transcendentals (2nd Edition)

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

Chapter 3 - Derivatives - 3.9 Derivatives of Logarithmic and Exponential Functions - 3.9 Exercises: 20

Answer

\[ - 2\tan x\]

Work Step by Step

\[\begin{gathered} \frac{d}{{dx}}\,\,\left[ {\ln \,\left( {{{\cos }^2}x} \right)} \right] \hfill \\ \hfill \\ Differentiate\,\,use\,\,\,the\,\,formula\,\,\frac{d}{{dx}}\,\,\left[ {\ln u} \right] = \frac{{{u^,}}}{u} \hfill \\ \hfill \\ let\,\,u = {\cos ^2}x\,\, \hfill \\ \hfill \\ then\,\,\,{u^,} = 2\cos x\,\left( { - \sin x} \right) \hfill \\ {u^,} = - 2\sin x\cos x \hfill \\ \hfill \\ substitute\,\,u{\text{ and }}{u^,} \hfill \\ \hfill \\ \frac{d}{{dx}}\,\,\left[ {\ln \,\left( {{{\cos }^2}x} \right)} \right] = \frac{{ - 2\sin x\cos x}}{{{{\cos }^2}x}} \hfill \\ \hfill \\ Simplify \hfill \\ \hfill \\ = \frac{{2\sin x}}{{\cos x}} = - 2\tan x \hfill \\ \hfill \\ \hfill \\ \end{gathered} \]
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