Differentiate `tan^(-1)((sqrt(1+x^2)-1)/x)` with respect to `sin^(-1)((2x)/(1+x^2)),` if `-1ltxlt1, x!=0.`

Explanation

`u=tan^(-1)((sqrt(1+x^2)-1)/x)`
`x=tantheta`
`u=tan^(-1)[(sqrt(1+tan^2theta)-1)/tantheta]`
`=tan^(-1)[(sectheta-1)/tantheta]`
`=tan^(-1)[(1-costheta)/sintheta]`
`u=tan^(-1)[tan(theta/2)]=theta/2`
`u=1/2tan^(-1)x`
`(du)/(dx)=d/dx[1/2tan^(-1)x]`
`v=sin^(-1)((2x)/(1+x^2))`
Let `x=tantheta`
`v=sin^(-1)[(2tantheta)/(1+tan^2theta)]`
`(dv)/dx=2/(1+x^2)`
`(du)/(dv)=(du)/dx*dx/(dv)`
`(du)/(dv)=1/4`.