University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 3 - Section 3.8 - Derivatives of Inverse Functions and Logarithms - Exercises - Page 174: 3


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Work Step by Step

$$f(x)=5-4x$$ a) To find $f^{-1}(x)$, we substitute $x$ with $f^{-1}x$ and $f(x)$ with $x$: $$x=5-4f^{-1}(x)$$ Then solve for $f^{-1}(x)$: $$4f^{-1}(x)=5-x$$ $$f^{-1}(x)=\frac{5-x}{4}$$ b) The graphs of $f(x)$ and $f^{-1}(x)$ are enclosed below. c) - Find $df/dx$: $$\frac{df}{dx}=\frac{d}{dx}\Big(5-4x\Big)=-4$$ At $x=1/2$, $df/dx=-4$ - Find $df^{-1}/dx$: $$\frac{df^{-1}}{dx}=\frac{d}{dx}\Big(\frac{5-x}{4}\Big)=\frac{1}{4}(0-1)=-\frac{1}{4}$$ At $x=f(1/2)=5-4\times(1/2)=3$, $$df^{-1}/dx=-\frac{1}{4}$$ It can be seen that at these points, $df^{-1}/dx=1/(df/dx)$
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