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: 11

Answer

\[{y^,} =- \frac{{15x}}{{3x + 2}} - 5\ln \,\left( {3x + 2} \right)\]

Work Step by Step

\[\begin{gathered} y = - 5x\ln \,\left( {3x + 2} \right) \hfill \\ Differentiate \hfill \\ {y^,} = \,\,\left[ { - 5x\ln \,\left( {3x + 2} \right)} \right] \hfill \\ Use\,\,the\,\,product\,\,rule \hfill \\ {y^,} = - 5x\,\,{\left[ {\ln \,\left( {3x + 2} \right)} \right]^,} + \ln \,\left( {3x + 2} \right)\,{\left( { - 5x} \right)^,} \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Then \hfill \\ {y^,} = - 5x\,\left( {\frac{{\,{{\left( {3x + 2} \right)}^,}}}{{3x + 2}}} \right) + \ln \,\left( {3x + 2} \right)\,\left( { - 5} \right) \hfill \\ {y^,} = - 5x\,\left( {\frac{3}{{3x + 2}}} \right) - 5\ln \,\left( {3x + 2} \right) \hfill \\ Multiply \hfill \\ {y^,} = -\frac{{15x}}{{3x + 2}} - 5\ln \,\left( {3x + 2} \right) \hfill \\ \end{gathered} \]
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