Differentiate `tan^(-1)((2x)/(1-x^2))` with respect to `sin^(-1)((2x)/(1+x^2))` , if `x in (1,\ oo)`

Explanation

Given that,
`u=tan^(-1)((2x)/(1-x^2))`
taking `x=tan theta`
`=>u=tan^-1((2tan theta)/(1-tan^2 theta))`
`=>u=tan^-1(tan2 theta)`
`v=sin^(-1)((2x)/(1+x^2))`
`=>v=sin^(-1)((2 tan theta)/(1+tan^2 theta))`
`=>v=sin^-1(sin 2 theta)`
`x in (1,oo)`
`=>tan theta in (1,oo)`
`=>theta in (pi/4,pi/2)`
`=>2 theta in (pi/2,pi)`
`u= 2 theta -pi`
`=>(du)/(d theta)=2`
`v=pi-2theta`
`=>(dv)/(d theta)=-2`
`=>(du)/(dv)=-1`