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 - Page 240: 13


\[{s^,} = t + 2t\ln \left| t \right|\,\]

Work Step by Step

\[\begin{gathered} s = {t^2}\ln \left| t \right| \hfill \\ Differentiate \hfill \\ {s^,} = \,\,{\left[ {{t^2}\ln \left| t \right|} \right]^,} \hfill \\ Use\,\,the\,\,product\,\,rule \hfill \\ {s^,} = {t^2}\,\,{\left[ {\ln \left| t \right|} \right]^,} + \ln \left| t \right|\,{\left( {{t^2}} \right)^,} \hfill \\ Use\,\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Then \hfill \\ {s^,} = {t^2}\,\,\left[ {\frac{{\,{{\left( t \right)}^,}}}{t}} \right] + \ln \,\left| t \right|\,\left( {2t} \right) \hfill \\ {s^,} = {t^2}\,\left( {\frac{1}{t}} \right) + \ln \left| t \right|\,\left( {2t} \right) \hfill \\ simplifying \hfill \\ {s^,} = t + 2t\ln \left| t \right|\, \hfill \\ \end{gathered} \]
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