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)