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: 61


$$\frac{d}{d x}(x f(x)) =x f^{\prime}(x)+f(x)$$

Work Step by Step

Since $$ f'(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ Then \begin{aligned} \frac{d}{d x}(x f(x)) &=\lim _{h \rightarrow 0} \frac{(x+h) f(x+h)-f(x)}{h}\\ &=\lim _{h \rightarrow 0} \frac{x f(x+h)+hf(x+h)-f(x)}{h}\\ &=\lim _{h \rightarrow 0}\left(x \frac{f(x+h)-f(x)}{h}+f(x+h)\right) \\ &=x \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}+\lim _{h \rightarrow 0} f(x+h) \\ &=x f^{\prime}(x)+f(x) \end{aligned}
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