If `f(x) = x tan^(-1) x`, find f'(1) from first principle.

Answer

`(1/2+pi/4)`

Explanation

Given, ` f(x) = x tan^(-1) x `
Using first principle,
`f'(1) = underset( h to 0) lim [(f(1+h)-f(1))/h]`
`=underset( h to 0) lim [((1+h) tan^(-1) (1+h) - tan^(-1) (1))/h]`
`=underset( h to 0) lim [(tan^(-1) (1+h) - tan^(-1)(1))/h+(h tan^(-1) (1+h))/h]`
` = underset( h to 0) lim [1/h tan^(-1) (h/(2+h)) + tan^(-1) (1+h)]`
`=underset( h to 0) lim [(tan^(-1)(h/(2+h)))/((2+h)*h/(2+h))]+pi/4`
`=underset( h to0) lim 1/(2+h) ((tan^(-1)(h/(2+h)))/(h/((2+h)))) + pi/4=1/2 + pi/4`