## Calculus (3rd Edition)

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