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: 16

Answer

\[{v^,} = \frac{{1 - 3\ln u}}{{{u^4}}}\]

Work Step by Step

\[\begin{gathered} V = \frac{{\ln u}}{{{u^3}}} \hfill \\ Use\,\,the\,\,quotient\,\,rule \hfill \\ {v^,} = \frac{{{u^3}\,\,{{\left[ {\ln u} \right]}^,} - \,\,\left[ {\ln u} \right]\,\,{{\left[ {{u^3}} \right]}^,}}}{{\,\,{{\left[ {{u^3}} \right]}^2}}} \hfill \\ Where \hfill \\ \ln u = \frac{{{u^,}}}{u} = \frac{1}{u} \hfill \\ Then \hfill \\ {v^,} = \frac{{{u^3}\,\left( {\frac{1}{u}} \right) - \ln u\,\left( {3{u^2}} \right)}}{{{u^6}}} \hfill \\ Simplify \hfill \\ {v^,} = \frac{{{u^2} - 3{u^2}\ln u}}{{{u^6}}} = \frac{{1 - 3\ln u}}{{{u^4}}} \hfill \\ \hfill \\ \hfill \\ \end{gathered} \]
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