Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

Published by Brooks Cole
ISBN 10: 1-43904-957-2
ISBN 13: 978-1-43904-957-0

Chapter 3 - Determining Change: Derivatives - 3.4 Activities - Page 224: 15

Answer

inside: $g(x)= x^{2}-3 x$ outside: $f(g)=g^{1/2}$ derivative: $f^{\prime}(x)=\dfrac{ 2x-3}{2\sqrt{x^2-3x}}$

Work Step by Step

Given $$ f(x)=\sqrt{x^{2}-3 x} $$ Rewriting $f(x) $ as$$ f(x)= (x^{2}-3 x)^{1/2} $$ Use the chain rule to take the derivative $$\frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x)$$ Here $g(x)=x^{2}-3 x$ and $ f(g )= g^{1/2}$, then \begin{align*} f'(x)&=( g^{1/2}(x))'\\ &=\frac{1}{2}g^{-1/2}(x)g'(x)\\ &=\frac{1}{2}(x^2-3x)^{-1/2} (2x-3)\\ &=\frac{2x-3}{2\sqrt{x^2-3x}} \end{align*}
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