Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.6 Trigonometric Functions - Exercises - Page 140: 41


$$ y''= 2\sec x (\sec x\tan x)=2\sec^2 x\tan x,$$ $$ y'''=4\sec^2 x\tan^2 x+2\sec^4x.$$

Work Step by Step

Since $ y=\tan x $, then we have $ y'=\sec^2 x $ and hence $$ y''= 2\sec x (\sec x\tan x)=2\sec^2 x\tan x,$$ and by using the product rule, we get $$ y'''=4\sec x(\sec x\tan x) \tan x+2 \sec^2 x\sec^2 x=4\sec^2 x\tan^2 x+2\sec^4x.$$
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