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: 34

Answer

\[{y^,} = \frac{1}{{\,\left( {\ln 3} \right)x}}\]

Work Step by Step

\[\begin{gathered} y = \log \,\left| {3x} \right| \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \,\,\,{\left[ {\log \,\left| {3x} \right|} \right]^,} \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{{\log }_a}\left| {g\,\left( x \right)} \right|} \right] = \frac{1}{{\ln a}} \cdot \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Here\,\,\,g\,\left( x \right) = 3x,\,\,a = 10 \hfill \\ Then \hfill \\ {y^,} = \,\frac{1}{{\ln 3}}\,\left( {\,\frac{{\,{{\left( {3x} \right)}^,}}}{{3x}}} \right) \hfill \\ {y^,} = \frac{1}{{\ln 3}}\,\left( {\frac{3}{{3x}}} \right) \hfill \\ Simplifying \hfill \\ {y^,} = \frac{1}{{\,\left( {\ln 3} \right)x}} \hfill \\ \end{gathered} \]
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