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: 25


inside: $g(x)=1+18 e^{0.6 x}$ outside: $f(g)= 12(g(x) )^{-1}$ derivative: $f^{\prime}(x)=-129.6e^{0.6x}(1+18 e^{0.6 x} )^{-2}$

Work Step by Step

Given$$ f(x)=\frac{12}{1+18 e^{0.6 x}} $$ Rewriting $f(x)$ as $$ f(x)= 12(1+18 e^{0.6 x})^{-1}$$ Use the chain rule to take the derivative $$ \frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x) $$ Here $g(x)=1+18 e^{0.6 x}$ and $f(g)= 12(g(x) )^{-1},$ then \begin{align*} f^{\prime}(x) &=\left(12(g(x) )^{-1}\right)^{\prime} \\ &=-12(g(x) )^{-2}g^{\prime}(x) \\ &=-12(1+18 e^{0.6 x} )^{-2}((0.6)18 e^{0.6 x}) \\ &=-129.6e^{0.6x}(1+18 e^{0.6 x} )^{-2} \end{align*}
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