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: 58


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

Work Step by Step

We evaluate the derivative as follows: \begin{aligned} \frac{d}{d x}\left[\frac{f(x)}{g(x)}\right] &=\frac{1}{h}\lim _{h \rightarrow 0} \frac{f(x+h)}{g(x+h)}-\frac{f(x)}{g(x)} \\ &=\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x+h)}{g(x+h) g(x) h} \\ &=\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x)+f(x) g(x)-f(x) g(x+h)}{g(x+h) g(x) h} \\ &=\lim _{h \rightarrow 0} \frac{[(x+h)-f(x)] g(x)-f(x)[(g(x+h)-g(x)]}{g(x+h) g(x) h} \\ &=\lim _{h \rightarrow 0} \frac{\frac{f(x+h)-f(x)}{h} g(x)-f(x) \frac{g(x+h)-g(x)}{h}}{g(x+h) g(x)}\\ &=\frac{ \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} g(x)-f(x) \lim _{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}}{ \lim _{h \rightarrow 0} g(x+h) g(x)}\\ &= \frac{f^{\prime}(x) g(x)-f(x) g^{\prime}(x)}{[g(x)]^{2}} \end{aligned}
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