Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 3: Derivatives - Section 3.1 - Tangents and the Derivative at a Point - Exercises 3.1 - Page 108: 21



Work Step by Step

To find the slope of a function at any given point, all one has to do would be to look for the value of the derivative function at the same x value. Given that $f(x)=\frac{1}{x-1}={(x-1)}^{-1}$, then the derivative function by the power rule will be $f'(x)=(-1)(x-1)^{-1-1}=-(x-1)^{-2}=\frac{-1}{(x-1)^2}$ As such, the slope at the point $x=3$ would be $f'(3)=\frac{-1}{(3-1)^2}=\frac{-1}{4}$
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