University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.8 - Derivatives of Inverse Functions and Logarithms - Exercises - Page 175: 57



Work Step by Step

$$y=\ln(3t e^{-t})$$ We have $$(\ln t)'=\frac{1}{t}$$ $$(e^{t})'=e^t$$ So the derivative of $y$ is $$y'=\frac{1}{3t e^{-t}}(3t e^{-t})'=\frac{3\Big(e^{-t}+t e^{-t}(-t)'\Big)}{3t e^{-t}}$$ $$y'=\frac{e^{-t}-t e^{-t}}{t e^{-t}}=\frac{e^{-t}(1-t)}{t e^{-t}}$$ $$y'=\frac{1-t}{t}$$
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