Q. 1 M

# Differentiate each of the following from first principles:

(x^{2} + 1)(x – 5)

Answer :

We need to find the derivative of f(x) = (x^{2} + 1)(x – 5)

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} + 1)(x – 5) is given as –

f’(x) =

⇒ f’(x) =

⇒ f’(x) =

Using (a + b)^{2} = a^{2} + 2ab + b^{2} and (a + b)^{3} = a^{3} + 3ab(a + b) + b^{3} we have –

⇒ f’(x) =

⇒ f’(x) =

Take h common –

⇒ f’(x) =

As h is cancelled, so there is no more indeterminate form possible if we put value of h = 0

∴ f’(x) =

So, evaluate the limit by putting h = 0

⇒ f’(x) = 3x^{2} + 3(0)x + 0^{2} + 1 – 10x – 5(0)

⇒ f’(x) = 3x^{2} – 10x + 1

Hence,

Derivative of f(x) = (x^{2} + 1)(x – 5) is 3x^{2} – 10x + 1

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