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: 81



Work Step by Step

$$y=\theta\sin(\log_7\theta)$$ The derivative of $y$ is $$y'=\sin(\log_7\theta)+\theta\cos(\log_7\theta)(\log_7\theta)'$$ Recall Theorem 7: $$\frac{d}{dx}\log_au=\frac{1}{u\ln a}\frac{du}{dx}$$ That means: $$(\log_7\theta)'=\frac{1}{\theta\ln7}(\theta)'=\frac{1}{\theta\ln7}$$ Therefore, $$y'=\sin(\log_7\theta)+\theta\cos(\log_7\theta)\times\frac{1}{\theta\ln7}$$ $$y'=\sin(\log_7\theta)+\frac{\cos(\log_7\theta)}{\ln7}$$ $$y'=\frac{\sin(\log_7\theta)\ln7+\cos(\log_7\theta)}{\ln7}$$
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