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: 5

Answer

$$\frac{dy}{dx}=\frac{e^\sqrt x}{2\sqrt x}$$

Work Step by Step

$$y=e^{\sqrt x}$$ $$\frac{dy}{dx}=\frac{d(e^{\sqrt x})}{dx}$$ Let $u=\sqrt x$ and $y=e^u$. Then, according to Chain Rule, $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$ $$\frac{dy}{dx}=\frac{d(e^u)}{du}\frac{d(\sqrt x)}{dx}$$ $$\frac{dy}{dx}=e^u\times\frac{d(x^{1/2})}{dx}$$ $$\frac{dy}{dx}=e^u\times\frac{1}{2}x^{-1/2}$$ $$\frac{dy}{dx}=e^{\sqrt x}\times\frac{1}{2\sqrt x}$$ $$\frac{dy}{dx}=\frac{e^\sqrt x}{2\sqrt x}$$
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