Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 2 - The Derivative - 2.5 Derivatives of Trigonometric Functions - Exercises Set 2.5: 24

Answer

$2sec^{2}(x)tan(x)$

Work Step by Step

The first derivative of $tan(x)$ is given as $sec^{2}(x)$ from the formula table. Writing $sec^{2}(x)$ as $sec(x) \times sec(x)$ allows the use of product rule to find the second derivative. With $f = sec(x)$ and $g=sec(x)$, $f'g+g'f$ (product rule) yields $sec(x)(sec(x)tan(x))$ + $sec(x)(sec(x)(tan(x)) = 2sec^{2}(x)tan(x)$.
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