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: 23

Answer

inside: $g(x)=0.08x$ outside: $f(g)=1+58e^{g}$ derivative: $f^{\prime}(x)=4.64 e^{0.08x}$

Work Step by Step

Given$$ f(x)=1+58 e^{0.08 x} $$ Use the chain rule to take the derivative $$ \frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x) $$ Here $g(x)=0.08x$ and $f(g)=1+58e^g,$ then \begin{align*} f^{\prime}(x) &=\left(1+58e^{g(x)}\right)^{\prime} \\ &=58e^{g(x)} g^{\prime}(x) \\ &=58(0.08)e^{0.08x}\\ &=4.64 e^{0.08x} \end{align*}
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