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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