Differentiate the following from first principles

tan 2x

We need to find derivative of f(x) = tan (2x)

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

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

∴ derivative of f(x) = tan (2x) is given as –

f’(x) =

⇒ 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) =

To apply sandwich theorem ,we need 2h in denominator, So multiplying by 2 in numerator and denominator by 2.

∴ f’(x) =

Use the formula –

⇒ f’(x) = 2×

∴ f’(x) =

Hence,

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