If `-1/(sqrt(3))ltxlt1/(sqrt(3))` , differentiate `tan^(-1)((3x-x^3)/(1-3x^2))` with respect to `tan^(-1)((2x)/(1-x^2))` .

Explanation

Given that,
`u=tan^(-1)((3x-x^3)/(1-3x^2))`
taking `x=tan theta`
`=>u=tan^-1((3tan theta-tan^3theta)/(1-3 tan^2 theta))`
`=>u=tan^-1(tan3theta)`
again `v=tan^(-1)((2x)/(1-x^2))`
`=>v=tan^(-1)((2tan theta)/(1-tan^2theta))`
`=>v=tan^-1(tan2theta)`
`-1/(sqrt(3))ltxlt1/(sqrt(3))`
`=>-1/(sqrt(3))lt tan theta lt1/(sqrt(3))`
`=>-pi/6 lt theta lt pi/6`
`=>-pi/3 lt 2theta lt pi/3`
`=>-pi/2 lt 3theta lt pi/2`
so,`u=3 theta`
`=>(du)/(d theta)=3`
`v=2 theta`
`=>(dv)/(d theta)=2`
`=>(du)/(dv)=3/2`