Q. 1 M

Differentiate each of the following from first principles:

(x2 + 1)(x – 5)

Answer :

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


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