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: 31

Answer

\[{y^,} = \frac{1}{{x\ln 6}}\]

Work Step by Step

\[\begin{gathered} y = \log \,\left( {6x} \right) \hfill \\ Use\,\,\,the\,\,formula\,\,in\,\,the\,\,page\,\,238 \hfill \\ \frac{d}{{dx}}\,\,\left[ {{{\log }_a}\left| {g\,\left( x \right)} \right|} \right] = \frac{1}{{\ln a}}\frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Then \hfill \\ {y^,} = \frac{1}{{\ln 6}}\,\,\left[ {\frac{{\,{{\left( {6x} \right)}^,}}}{{6x}}} \right] \hfill \\ Then \hfill \\ {y^,} = \frac{1}{{\ln 6}}\,\left( {\frac{6}{{6x}}} \right) \hfill \\ {y^,} = \frac{1}{{x\ln 6}} \hfill \\ \end{gathered} \]
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