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# Differentiate each of the following from first principles:

(x + 2)^{3}

Answer :

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 a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})

⇒ 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}

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