Differentiate the following from first principles
We need to find derivative of f(x) = x2 sin x
Derivative of a function f(x) is given by –
f’(x) = {where h is a very small positive number}
∴ derivative of f(x) = x2 sin x is given as –
f’(x) =
⇒ f’(x) =
Using (a + b)2 = a2 + 2ab + b2 ,we have –
⇒ f’(x) =
Using algebra of limits, we have –
⇒ f’(x) =
⇒ f’(x) =
⇒ f’(x) = 0×sin (x + 0) + 2x sin (x + 0) +
⇒ f’(x) =
Using algebra of limits we have –
∴ f’(x) = 2x sin x +
We can’t evaluate the limits at this stage only as on putting value it will take 0/0 form. So, we need to do little modifications.
Use: sin A – sin B = 2 cos ((A + B)/2) sin ((A – B)/2)
∴ f’(x) =
⇒ f’(x) =
Using algebra of limits –
⇒ f’(x) =
Use the formula –
∴ f’(x) =
Put the value of h to evaluate the limit –
∴ f’(x) = 2x sin x + x2 cos(x + 0) = 2x sin x + x2 cos x
Hence,
Derivative of f(x) = (x2 sin x) is (2x sin x + x2 cos x)