Q. 2 J5.0( 2 Votes )

# Differentiate the following from first principle.

x cos x

Answer :

We need to find derivative of f(x) = x cos x

Derivative of a function f(x) is given by –

f’(x) = {where h is a very small positive number}

∴ derivative of f(x) = x cos x is given as –

f’(x) =

⇒ f’(x) =

⇒ f’(x) =

Using algebra of limits, we have –

⇒ f’(x) =

⇒ f’(x) =

Using algebra of limits we have –

∴ f’(x) = cos 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: cos A – cos B = – 2 sin ((A + B)/2) sin ((A – B)/2)

∴ f’(x) = cos x +

⇒ f’(x) = cos x –

Using algebra of limits –

⇒ f’(x) =

Use the formula –

∴ f’(x) =

Put the value of h to evaluate the limit –

∴ f’(x) = cos x – x sin x

Hence,

Derivative of f(x) = x cos x is cos x – x sin x

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