Calculus with Applications (10th Edition)

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

Chapter 4 - Calculating the Derivative - 4.4 Derivatives of Exponential Functions - 4.4 Exercises - Page 232: 27


$6x(\ln 4)4^{x^{2}+2}$

Work Step by Step

$y=3\cdot 4^{x^{2}+2}$ $y=3g(x),\qquad g(x)=4^{x^{2}+2}$, $\displaystyle \frac{dy}{dx}=3\cdot\frac{d}{dx}[g(x)],$ g(x) is a composite function, $g(x)=4^{h(x)},\qquad h(x)=x^{2}+2, \quad h^{\prime}(x)=2x$ Using$\quad \color{blue}{ \displaystyle \frac{d}{dx}(a^{h(x)})=(\ln a)a^{h(x)}\cdot h^{\prime}(x)}$, $\displaystyle \frac{dy}{dx}=3\cdot\ln 4\cdot 4^{x^{2}+2}\cdot 2x$ $\ \ \ \ \ \ =6x\cdot\ln 4\cdot 4^{x^{2}+2}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.