Differentiate `cos^(-1)((2x)/(1+x^2))` , `-1ltxlt1` with respect to`x`

Explanation

Given that,
`y=cos^(-1)((2x)/(1+x^2))`
`-1ltxlt1`
`x= tan theta`
so, `y=cos^-1 [(2tantheta)/(1+tan^2 theta)]`
`=>y=cos^-1 (sin 2theta)`
`=>y=cos^-1 [cos(pi/2-2theta)]`
`-1ltxlt1`
`=> -1lt tan theta lt1`
`=>-pi/4 lt theta lt pi/4`
`=>-pi/2 lt 2 theta lt pi/2`
`=>0 lt pi/2 - 2theta lt pi`
`y=pi/2 -2theta`
`=>y=pi/2 tan^-1 x`
`(dy)/(dx)=-2/(1+x^2)`