We need to find derivative of f(x) = tan x²

Derivative of a function f(x) from first principle is given by –

f’(x) = {where h is a very small positive number}

∴ derivative of f(x) = tan x²is given as –

f’(x) =

⇒ f’(x) =

⇒ f’(x) =

⇒ f’(x) =

Use the formula: sin (A – B) = sin A cos B – cos A sin B

⇒ f’(x) =

Using algebra of limits, we have –

⇒ f’(x) =

⇒ f’(x) =

⇒ f’(x) =

As, h → 0 ⇒ 2hx + h² → 0

∴ To use the sandwich theorem to evaluate the limit, we need 2hx + h² in denominator. So multiplying this in numerator and denominator.

⇒ f’(x) =

Using algebra of limits –

⇒ f’(x) =

⇒ f’(x) =

Use the formula:

∴ f’(x) = sec² x² × 1 × (2x + 0)

∴ f’(x) = 2x sec² x²

Hence,

Derivative of f(x) = tan x² = 2x sec² x²