Correct Answer  Option 1 : 2 tan x sec
2 x
Concept:
Chain Rule of Derivatives:

\(\rm \frac{d}{dx}f(g(x))=\frac{d}{d\ g(x)}f(g(x))\times \frac{d}{dx}g(x)\).

\(\rm \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\).
Derivatives of Trigonometric Functions:
\(\rm \frac{d}{dx}\sin x=\cos x\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\cos x=\sin x\\ \frac{d}{dx}\tan x=\sec^2x\ \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\cot x=\csc^2 x\\ \frac{d}{dx}\sec x=\tan x\sec x\ \ \ \ \frac{d}{dx}\csc x=\cot x\csc x\)
Calculation:
Using the chain rule of derivatives, we get:
\(\rm \frac{d}{dx}\tan^2x\) = \(\rm \frac{d \tan^2x}{d \tan x} \times \frac {d\tan x}{dx}\)
= 2 tan x sec^{2} x.