Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Practice Exercises - Page 439: 26


$$\frac{1}{10}\sqrt[10]{\frac{3 x+4}{2 x-4}} \left( \frac{3}{ 3 x+4}-\frac{ 2}{2 x-4}\right)$$

Work Step by Step

Given $$ y=\sqrt[10]{\frac{3 x+4}{2 x-4}} $$ Since \begin{align*} \ln y&= \frac{1}{10} \ln \frac{3 x+4}{2 x-4}\\ &= \frac{1}{10}\left( \ln (3 x+4)-\ln (2 x-4)\right)\\ \end{align*} Differentiate both sides; we get \begin{align*} \frac{y'}{y}&= \frac{1}{10}\left( \frac{3}{ 3 x+4}-\frac{ 2}{2 x-4}\right)\\ y'&= \frac{y}{10}\left( \frac{3}{ 3 x+4}-\frac{ 2}{2 x-4}\right)\\ &=\frac{1}{10}\sqrt[10]{\frac{3 x+4}{2 x-4}} \left( \frac{3}{ 3 x+4}-\frac{ 2}{2 x-4}\right) \end{align*}
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