University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.6 - The Chain Rule - Exercises - Page 160: 108

Answer

See below for proof that the Power Rule holds for the function $x^{3/4}$.

Work Step by Step

$$x^{3/4}=\sqrt{x\sqrt x}=(x\times x^{1/2})^{1/2}=(x^{3/2})^{1/2}$$ Using the Chain Rule, we have $$\frac{d}{dx}(x^{3/2})^{1/2}=\frac{1}{2}(x^{3/2})^{-1/2}(x^{3/2})'=\frac{1}{2}(x^{3/2})^{-1/2}\times\frac{3}{2}x^{1/2}$$ $$=\frac{3}{4}x^{-3/4}\times x^{1/2}=\frac{3}{4}x^{-1/4}=\frac{3}{4}x^{3/4-1}$$ So the Power Rule $(x^n)'=nx^{n-1}$ holds for the function $x^{3/4}$.
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