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: 59


$$\frac{d}{d x}\left(\frac{1}{f(x)}\right) =-\frac{f^{\prime}(x)}{f^{2}(x)}$$

Work Step by Step

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