Differentiate each of the following from first principles:
(x + 2)3
We need to find the derivative of f(x) = (x + 2)3
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) = (x + 2)3 is given as –
f’(x) =
⇒ f’(x) =
Using a3 – b3 = (a – b)(a2 + ab + b2)
⇒ f’(x) =
⇒ f’(x) =
As h is cancelled, so there is no more indeterminate form possible if we put value of h = 0.
So, evaluate the limit by putting h = 0
⇒ f’(x) =
⇒ f’(x) = (x + 0 + 2)2 + (x + 2)(x + 2) + (x + 2)2
⇒ f’(x) = 3 (x + 2)2
⇒ f’(x) = 3 (x + 2)2
Hence,
Derivative of f(x) = (x + 2)3 is 3(x + 2)2