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: 6

Answer

\[{y^,} = \frac{{2 - 24{x^2}}}{{2x - 8{x^3}}}\]

Work Step by Step

\[\begin{gathered} y = \ln \left| { - 8{x^3} + 2x} \right| \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \,\,\left[ {\ln \,\left| { - 8{x^3} + 2x} \right|} \right] \hfill \\ Use\,\,the\,\,formula\,\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Here\,\,g\,\left( x \right) = - 8{x^3} + 2x \hfill \\ Then \hfill \\ {y^,} = \frac{{\,{{\left( { - 8{x^3} + 2x} \right)}^,}}}{{ - 8{x^3} + 2x}} \hfill \\ Differentiating \hfill \\ {y^,} = \frac{{ - 24{x^2} + 2}}{{ - 8{x^3} + 2x}} \hfill \\ {y^,} = \frac{{2 - 24{x^2}}}{{2x - 8{x^3}}} \hfill \\ \end{gathered} \]
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