Calculus: Early Transcendentals 8th Edition

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

Chapter 3 - Section 3.1 - Derivatives of Polynomials and Exponential Functions - 3.1 Exercises: 47


$f'(x)=2-\frac{15}{4}x^{-1/4}$ $f''(x)=\frac{15}{16}x^{-5/4}$ These functions are reasonable since f' is positive when f is increasing and negative when f is decreasing. The same goes for f'' and f' - since f' is always increasing, f'' is always positive. The slopes of the derivatives are closer to zero wherever the graphs of the original functions appear almost flat.

Work Step by Step

$f(x)=2x-5x^{3/4}$ Use the Power Rule: $f(x) = x^{n},$ then $f'(x) = nx^{n-1}$ $f'(x)=2(1)-5(3/4)x^{-1/4}$ $f'(x)=2-\frac{15}{4}x^{-1/4}$ $f''(x)=0-(\frac{15}{4})(-\frac{1}{4})x^{-5/4}$ $f''(x)=\frac{15}{16}x^{-5/4}$
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