Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - Chapter 6 Review Exercises - Page 485: 24

Answer

$$y' = \frac{1}{{3\left( {x + 1} \right)}}$$

Work Step by Step

$$\eqalign{ & y = \ln \left( {\root 3 \of {x + 1} } \right) \cr & or \cr & y = \ln {\left( {x + 1} \right)^{1/3}} \cr & {\text{logarithmic property }}\ln {u^v} = v\ln u \cr & y = \frac{1}{3}\ln \left( {x + 1} \right) \cr & {\text{find the derivative}} \cr & y' = \left( {\frac{1}{3}\ln \left( {x + 1} \right)} \right)' \cr & y' = \frac{1}{3}\left( {\ln \left( {x + 1} \right)} \right)' \cr & {\text{use }}\left( {\ln u} \right)' = \frac{{u'}}{u}, \cr & y' = \frac{1}{3}\left( {\frac{{\left( {x + 1} \right)'}}{{x + 1}}} \right) \cr & y' = \frac{1}{3}\left( {\frac{1}{{x + 1}}} \right) \cr & {\text{simplifying}} \cr & y' = \frac{1}{{3\left( {x + 1} \right)}} \cr} $$
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