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: 61


$$y'=e^{\cos t+\ln t}\Big(\frac{1}{t}-\sin t\Big)$$

Work Step by Step

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