Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.7 The Chain Rule - 3.7 Exercises: 36

Answer

a. $\frac{d}{dx}(e^x)^3 = 3e^{3x}$ b. $\frac{d}{dx}e^{x^3} = 3x^2e^{x^3}$

Work Step by Step

Chain Rule $\frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x)$ a. $y=(e^x)^3$ Outer Function: $y=f(u)=u^3$ Inner Function: $u=g(x)=e^x$ $\frac{d}{dx}(e^x)^3 = 3e^{2x} \times e^x = 3e^{3x}$ b. $y=e^{x^3}$ Outer Function: $y=f(u)=e^x$ Inner Function: $u=g(x)=x^3$ $\frac{d}{dx}e^{x^3} = e^{x^3} \times 3x^2$
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