Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - Review - Exercises - Page 198: 69


$$h'(x)=\frac{f^{\prime}(x)[g(x)]^{2}+g^{\prime}(x)[f(x)]^{2}}{[f(x)+g(x)]^{2}} $$

Work Step by Step

Given $$ h(x)=\frac{f(x)g(x)}{f(x)+g(x)}$$ Then $\begin{align*} h^{\prime}(x) &=\frac{[f(x)+g(x)]\left[f(x) g^{\prime}(x)+g(x) f^{\prime}(x)\right]-f(x) g(x)\left[f^{\prime}(x)+g^{\prime}(x)\right]}{[f(x)+g(x)]^{2}} \\ &=\frac{[f(x)]^{2} g^{\prime}(x)+f(x) g(x) f^{\prime}(x)+f(x) g(x) g^{\prime}(x)+[g(x)]^{2} f^{\prime}(x)}{[f(x)+g(x)]^{2}}-\frac{f(x) g(x) f^{\prime}(x)+f(x) g(x) g^{\prime}(x)}{[f(x)+g(x)]^{2}} \\ &=\frac{f^{\prime}(x)[g(x)]^{2}+g^{\prime}(x)[f(x)]^{2}}{[f(x)+g(x)]^{2}} \end{align*}$
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