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