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: 1

Answer

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

Work Step by Step

\[\begin{gathered} y = \ln \,\left( {8x} \right) \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \,\,{\left[ {\ln \,\left( {8x} \right)} \right]^,} \hfill \\ Use\,\,the\,\,formula\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Here\,\,g\,\left( x \right) = 8x\, \hfill \\ Then \hfill \\ {y^,} = \frac{{\,{{\left( {8x} \right)}^,}}}{{8x}} \hfill \\ {y^,} = \frac{8}{{8x}} \hfill \\ Simplifying \hfill \\ {y^,} = \frac{1}{x} \hfill \\ \hfill \\ \end{gathered} \]
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