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

Explanation

Let ` y = tan^(-1) ((sqrt(1 + x^(2))-1)/(x))and v = tan ^(-1) x `
On putting ` x = tan theta ` , we get
` y = tan^(-1) ((sqrt(1 + x^(2))-1)/(x))= tan ^(-1) ((1 -cos theta )/(sin theta ))` ltrgt ` rArr u = tan^(-1)[(2 sin ^(2) (theta //2))/(2 sin (theta//2) cos (theta //2))] = tan^(-1) (tan(theta)/(2))`
` = (1)/(2) theta = (1)/(2) tan^(-1) x `
Thus , we have ` u = (1)/(2) tan^(-1) x and v tan^(-1) x `
On differentiating both functions w.r.t.x, we get
` (du)/(dx) = (1)/(2)xx(1)/(1 + x^(2))`
and `(dv)/(dx) = (1)/(1 + x^(2))`
` therefore (du)/(dv) = (du//dx)/(dv//dx) = (1)/(2(1 + x^(2))) xx (1 + x^(2)) = (1)/(2)`