Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 386: 57


$$\frac{e^{3 x}(x-2)^{2}}{(x+1)^{2}} \left(3+\frac{2}{ x-2} -\frac{2}{ x+1}\right)$$

Work Step by Step

Given $$y=\frac{e^{3 x}(x-2)^{2}}{(x+1)^{2}}$$ Since \begin{align*} \ln y&=\ln \frac{e^{3 x}(x-2)^{2}}{(x+1)^{2}}\\ &=\ln( e^{3 x}(x-2)^{2})-\ln((x+1)^{2})\\ &= \ln( e^{3 x} )+2\ln( x -2 )-2\ln( x+1 ) \end{align*} Differentiate both sides \begin{align*} \frac{1}{y}y'&= \frac{1}{e^{3x}}3e^{3x}+\frac{2}{ x-2} -\frac{2}{ x+1} \\ &= 3+\frac{2}{ x-2} -\frac{2}{ x+1} \\ y'&=\frac{e^{3 x}(x-2)^{2}}{(x+1)^{2}} \left(3+\frac{2}{ x-2} -\frac{2}{ x+1}\right) \end{align*}
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