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: 4

Answer

\[{y^,} = \frac{{3{x^2}}}{{1 + {x^3}}}\]

Work Step by Step

\[\begin{gathered} y = \ln \,\left( {1 + {x^3}} \right) \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \,\,\left[ {\ln \,\left( {1 + {x^3}} \right)} \right] \hfill \\ Use\,\,the\,\,formula\,\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ Here\,\,g\,\left( x \right) = 1 + {x^3}\, \hfill \\ Then \hfill \\ {y^,} = \frac{{\,{{\left( {1 + {x^3}} \right)}^,}}}{{1 + {x^3}}} \hfill \\ Differentiating \hfill \\ {y^,} = \frac{{3{x^2}}}{{1 + {x^3}}} \hfill \\ \end{gathered} \]
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