Differentiate the following from first principles
We need to find derivative of f(x) =
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) = is given as –
f’(x) =
⇒ f’(x) =
Use: sin A – sin B = 2 cos ((A + B)/2) sin ((A – B)/2)
⇒ f’(x) =
Using algebra of limits, we have –
⇒ f’(x) = 2
⇒ f’(x) = 2
⇒ f’(x) =
As, h → 0 ⇒ → 0
∴ To use the sandwich theorem to evaluate the limit, we need in denominator. So multiplying this in numerator and denominator.
⇒ f’(x) =
Using algebra of limits –
⇒ f’(x) =
Use the formula:
∴ f’(x) = × 1 ×
⇒ f’(x) =
⇒ f’(x) =
Again we get an indeterminate form, so multiplying and dividing √(2x + 2h) + √(2x) to get rid of indeterminate form.
∴ f’(x) =
Using a2 – b2 = (a + b)(a – b), we have –
⇒ f’(x) =
Using algebra of limits we have –
⇒ f’(x) =
⇒ f’(x) =
⇒ f’(x) =
∴ f’(x) =
Hence,
Derivative of f(x) = sin √2x =