Differentiate `tan^(-1)((sqrt(1+x^2-1))/x)` with respect to `tan^(-1)x ,` when `x!=0.`

Explanation

Let `u = tan^(-1)'(sqrt(1+x^(2))-1)/(x)` and `v = tan^(-1)x`
`:. x = tan theta`
`rArr u = tan^(-1)'(sqrt(1+tan^(2)theta)-1)/(tan theta)`
` = tan^(-1)'((sectheta - 1)costheta)/(sintheta)`
`=tan^(-1)((1-cos theta)/(sintheta))`
`= tan^(-1) [(1-1+2sin^(2)theta//2)/(2sintheta//2.costheta//2)], [:' cos theta = 1-2sin^(2) theta]`
`= tan^(-1)[tan'(theta)/(2)]`
`= (theta)/(2) = (1)/(2) tan^(-1)x`
`:. (du)/(dx) = 1/2 (d)/(dx) tan^(-1)x = (1)/(2).(1)/(1+x^(2))"..."(i)`
and `(dv)/(dx) = (d)/(dx) tan^(-1)x= (1)/(1+x^(2))"...."(ii)`
` :. (du)/(dv) = (du//dx)/(dv//dx)`
`= (1//2(1+x^(2)))/(1//(1+x^(2))) = ((1+x^(2)))/(2(1+x^(2))) = (1)/(2)`