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