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