Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.4 - The Chain Rule - 3.4 Exercises: 2

Answer

$$\frac{dy}{dx}=24x^2(2x^3+5)^3$$

Work Step by Step

$$y=(2x^3+5)^4$$ $$\frac{dy}{dx}=\frac{d(2x^3+5)^4}{dx}$$ Let $u=2x^3+5$ and $y=u^4$. Then, according to Chain Rule, $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ $$\frac{dy}{dx}=\frac{d(u^4)}{du}\frac{d(2x^3+5)}{dx}$$ $$\frac{dy}{dx}=4u^3\times(6x^2+0)$$ $$\frac{dy}{dx}=4(2x^3+5)^3\times6x^2$$ $$\frac{dy}{dx}=24x^2(2x^3+5)^3$$
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