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The derivative, \(\rm \frac{d\tan^2x}{dx}\), is equal to:
1. 2 tan x sec2 x
2. 2 tan x sec x
3. 2 tan2 x sec2 x
4. None of these.

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Correct Answer - Option 1 : 2 tan x sec2 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 sec2 x.

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