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)