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: 10

Answer

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

Work Step by Step

\[\begin{gathered} y = \ln \,{\left( {5{x^3} - 2x} \right)^{3/2}} \hfill \\ Use\,\,the\,\,\log \,\,property \hfill \\ \ln {u^n} = n\ln u\, \hfill \\ y = \frac{3}{2}\ln \left( {5{x^3} - 2x} \right) \hfill \\ Differentiate \hfill \\ {y^,} = \,\,{\left[ {\frac{3}{2}\ln \left( {5{x^3} - 2x} \right)} \right]^,} \hfill \\ Pull\,\,out\,\,the\,\,constant \hfill \\ {y^,} = \frac{3}{2}\,\,\left[ {\ln \left( {5{x^3} - 2x} \right)} \right] \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Then \hfill \\ {y^,} = \frac{3}{2}\,\left( {\frac{{\,{{\left( {5{x^3} - 2x} \right)}^,}}}{{5{x^3} - 2x}}} \right) \hfill \\ {y^,} = \frac{3}{2}\,\left( {\frac{{15{x^2} - 2}}{{5{x^3} - 2x}}} \right) \hfill \\ \hfill \\ \end{gathered} \]
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