Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.3 Product and Quotient Rules - Exercises - Page 123: 60


$$\frac{d}{d x}\left(\frac{g(x)}{f(x)}\right) =\frac{f(x) g^{\prime}(x)-g(x) f^{\prime}(x)}{f^{2}(x)}$$

Work Step by Step

Since we have $$\frac{d}{d x}\left(\frac{1}{f(x)}\right)=-\frac{f^{\prime}(x)}{f^{2}(x)}\tag{1}$$ Then \begin{aligned} \frac{d}{d x}\left(\frac{g(x)}{f(x)}\right)&=g(x)\left(\frac{1}{f(x)}\right)^{\prime}+\frac{1}{f(x)}(g(x))^{\prime},\ \ \text{Use }\ \ (1) \\ &=g(x)\left(\frac{-f^{\prime}(x)}{f^{2}(x)}\right)+\frac{(g(x))^{\prime}}{f(x)}\\ &=\frac{-g(x) f^{\prime}(x)}{f^{2}(x)}+\frac{f(x)\left(g^{\prime}(x)\right)}{f^{2}(x)}\\ & =\frac{f(x) g^{\prime}(x)-g(x) f^{\prime}(x)}{f^{2}(x)} \end{aligned}
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