Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 386: 34


$$ f'(x) =\frac{e^{\sec^{-1}x}}{|x|\sqrt{x^2-1}} .$$

Work Step by Step

Recall that $(e^x)'=e^x$ Recall that $(\sec^{-1} x)'=\dfrac{1}{|x|\sqrt{x^2-1}}$ Since $ f(x)=e^{\sec^{-1}x}$, then the derivative, using the chain rule, is given by $$ f'(x)=e^{\sec^{-1}x}(\sec^{-1}x)'=e^{\sec^{-1}x}\frac{1}{|x|\sqrt{x^2-1}} \\ =\frac{e^{\sec^{-1}x}}{|x|\sqrt{x^2-1}} .$$
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