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: 15

Answer

\[{y^,} = \frac{{\frac{{2{x^2}}}{{x + 3}} - 4x\ln \,\left( {x + 3} \right)}}{{{x^4}}}\]

Work Step by Step

\[\begin{gathered} y = \frac{{2\ln \,\left( {x + 3} \right)}}{{{x^2}}} \hfill \\ Differentiate \hfill \\ {y^,} = \,\,\,\,{\left[ {\frac{{2\ln \,\left( {x + 3} \right)}}{{{x^2}}}} \right]^,} \hfill \\ Use\,\,the\,\,quotient\,\,rule \hfill \\ {y^,} = \frac{{{x^2}\,{{\left( {2\ln \,\left( {x + 3} \right)} \right)}^,} - 2\ln \,\left( {x + 3} \right)\,{{\left( {{x^2}} \right)}^,}}}{{\,{{\left( {{x^2}} \right)}^2}}} \hfill \\ Then \hfill \\ {y^,} = \frac{{{x^2}\,\left( {\frac{2}{{x + 3}}} \right) - 2\ln \,\left( {x + 3} \right)\,\left( {2x} \right)}}{{{x^4}}} \hfill \\ Simplify \hfill \\ {y^,} = \frac{{\frac{{2{x^2}}}{{x + 3}} - 4x\ln \,\left( {x + 3} \right)}}{{{x^4}}} \hfill \\ \end{gathered} \]
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