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: 22


inside: $g(x)=4 x^2$ outside: $f(g)=e^{g}$ derivative: $f^{\prime}(x)=8xe^{4x^2}$

Work Step by Step

Given$$ f(x)=e^{4 x^2} $$ Use the chain rule to take the derivative $$ \frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x) $$ Here $g(x)=4x^2$ and $f(g)=e^g,$ then \begin{align*} f^{\prime}(x) &=\left(e^{g(x)}\right)^{\prime} \\ &=e^{g(x)} g^{\prime}(x) \\ &=8xe^{4x^2} \end{align*}
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