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: 28


inside: $g(x)=2^x$ outside: $f(g)= \ln(g(x))$ derivative: $f^{\prime}(x)=\dfrac{2^x\ln 2}{2^x}=\ln 2 $

Work Step by Step

Given$$ f(x)=\ln(2^x) $$ Use the chain rule to take the derivative $$ \frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x) $$ Here $g(x)= 2^x$ and $f(g)= \ln(g(x)),$ then \begin{align*} f^{\prime}(x) &=\left( \ln(g(x))\right)^{\prime} \\ &=\frac{1}{g(x)}g^{\prime}(x) \\ &=\frac{2^x\ln 2}{2^x}\\ &=\ln 2 \end{align*}
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