Differentiate the following from first principles

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

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

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

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

f’(x) =

⇒ f’(x) =

As the limit takes 0/0 form on putting h = 0. So we need to remove the indeterminate form. As the numerator expression has square root terms so we need to multiply numerator and denominator by √tan (x + h) + √tan x.

⇒ f’(x) =

Using (a + b)(a – b) = a² – b² ,we have –

⇒ f’(x) =

Using algebra of limits, we have –

⇒ f’(x) =

⇒ f’(x) =

⇒ f’(x) =

⇒ f’(x) =

Using: sin A cos B – cos A sin B = sin (A – B)

⇒ f’(x) =

Using algebra of limits we have –

∴ f’(x) =

Use the formula –

⇒ f’(x) = ×

∴ f’(x) =

Hence,

Derivative of f(x) = √tan(x) is