If `f(x)=xtan^(-1)x ,` find `f^(prime)(sqrt(3))` using the first principle.

Explanation

We have
`f'(x)=underset(hrarr0)lim(f(x+h)-f(x))/(h)`
`therefore" "f'(sqrt(3))=underset(hrarr0)lim(f(sqrt(3)+h)-f(sqrt(3)))/(h)`
`=underset(hrarr0)lim((sqrt(3)+h)tan^(-1)(sqrt(3)+h)-sqrt(3)tan^(-1)sqrt(3))/(h)`
`=underset(hrarr0)lim(sqrt(3))/(h)tan^(-1)((sqrt(3+h)-sqrt(3))/(1+sqrt(3)(sqrt(3)+h)))+underset(hrarr0)limtan^(-1)(sqrt(3)+h)`
`=sqrt(3)underset(hrarr0)lim{tan^(-1)((h)/(4+sqrt(3)h))/((h)/(4+sqrt(3)h))}(1)/(4+sqrt(3)h)+underset(hrarr0)limtan^(-1)(sqrt(3)+h)`
`=sqrt(3)xx1xx(1)/(4)+tan^(-1)sqrt(3)=(sqrt(3))/(4)+tan^(-1)sqrt(3)`