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: 7

Answer

\[{y^,} = \frac{1}{{2\,\left( {x + 5} \right)}}\]

Work Step by Step

\[\begin{gathered} y = \ln \sqrt {x + 5} \hfill \\ Write\,\,\sqrt {x + 5} \,\,as\,\,\,{\left( {x + 5} \right)^{1/2}} \hfill \\ y = \ln \,\,\,{\left( {x + 5} \right)^{1/2}} \hfill \\ Use\,\,the\,\,\log \,\,property \hfill \\ \ln {a^n} = n\ln a \hfill \\ y = \frac{1}{2}\ln \,\left( {x + 5} \right) \hfill \\ Differentiating \hfill \\ {y^,} = \,\,{\left[ {\frac{1}{2}\ln \,\left( {x + 5} \right)} \right]^,} \hfill \\ {y^,} = \frac{1}{2}\,\,{\left[ {\ln \,\left( {x + 5} \right)} \right]^,} \hfill \\ Use\,\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ {y^,} = \frac{1}{2}\,\left( {\frac{1}{{x + 5}}} \right) \hfill \\ {y^,} = \frac{1}{{2\,\left( {x + 5} \right)}} \hfill \\ \hfill \\ \end{gathered} \]
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