Differentiate each of the following from first principles:
(x2 + 1)(x – 5)
We need to find the derivative of f(x) = (x2 + 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) = (x2 + 1)(x – 5) is given as –
f’(x) =
⇒ f’(x) =
⇒ f’(x) =
Using (a + b)2 = a2 + 2ab + b2 and (a + b)3 = a3 + 3ab(a + b) + b3 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) = 3x2 + 3(0)x + 02 + 1 – 10x – 5(0)
⇒ f’(x) = 3x2 – 10x + 1
Hence,
Derivative of f(x) = (x2 + 1)(x – 5) is 3x2 – 10x + 1