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: 35

Answer

\[{y^,} = \frac{5}{{2\ln 5\,\left( {5x + 2} \right)}}\]

Work Step by Step

\[\begin{gathered} y = {\log _5}\sqrt {5x + 2} \hfill \\ Write\,\,\sqrt {5x + 2} \,\,as\,\,\,{\left( {5x + 2} \right)^{1/2}} \hfill \\ y = {\log _5}\,{\left( {5x + 2} \right)^{1/2}} \hfill \\ Use\,\,\log \,\,property \hfill \\ y = \frac{1}{2}{\log _5}\,\left( {5x + 2} \right) \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \frac{1}{2}\,\,{\left[ {{{\log }_5}\,\left( {5x + 2} \right)} \right]^,} \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{{\log }_a}\left| {g\,\left( x \right)} \right|} \right] = \frac{1}{{\ln a}} \cdot \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Then \hfill \\ {y^,} = \frac{1}{{2\ln 5}}\,\left( {\frac{5}{{5x + 2}}} \right) \hfill \\ {y^,} = \frac{5}{{2\ln 5\,\left( {5x + 2} \right)}} \hfill \\ \end{gathered} \]
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