Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.5 Derivatives of Logarithmic Functions - 4.5 Exercises: 23

Answer

\[{y^,} = \frac{1}{{x\ln x}}\]

Work Step by Step

\[\begin{gathered} y = \ln \left| {\ln x} \right| \hfill \\ Differentiate \hfill \\ {y^,} = \,\,{\left[ {\ln \left| {\ln x} \right|} \right]^,} \hfill \\ Use\,\,the\,\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Here\,\,g\,\left( x \right) = \ln x \hfill \\ {y^,} = \frac{{\,\,{{\left[ {\ln x} \right]}^,}}}{{\ln x}} \hfill \\ Then \hfill \\ {y^,} = \frac{{1/x}}{{\ln x}} \hfill \\ {y^,} = \frac{1}{{x\ln x}} \hfill \\ \end{gathered} \]
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