Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.3 The Chain Rule - 4.3 Exercises - Page 225: 42


Please see explanation in "step-by-step"

Work Step by Step

Let $f(x)=x^{n}$. Then, $y=f[g(x)]=[g(x)]^{n}\quad $ is a composite function. To find $\displaystyle \frac{dy}{dx}$, we use the chain rule, $\displaystyle \frac{dy}{dx}=f^{\prime}[g(x)]\cdot g^{\prime}(x)$. By the power rule$, f^{\prime}(x)=nx^{n-1}$, so $\displaystyle \frac{dy}{dx}=n[g(x)]^{n-1}\cdot g^{\prime}(x)$.
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