Differentiate `cos^(-1)(2xsqrt(1-x^2)) ` , `-1/(sqrt(2)) lt x lt 1/sqrt2`

Explanation

`y=cos^(-1)(2xsqrt(1-x^2))`
Putting `x=sintheta`
`y=cos^(-1)(2sinthetasqrt(1-sin^2theta))`
`y=cos^(-1)(2sinthetasqrt(cos^2theta))`
`y=cos^(-1)(2sinthetacostheta)`
`y=cos^(-1)(sin2theta)`
`y=cos^(-1)[cos(pi/2-2theta)]`
`-1/(sqrt(2)) lt x lt 1/sqrt2`
`=>-1/(sqrt(2)) lt sin theta lt 1/sqrt2`
`=>-pi/4 lt theta lt pi/4`
`=>-pi/2 lt 2theta lt pi/2`
`=>0 lt pi/2 -2theta lt pi`
`y= pi/2 - 2theta`
`=>y= pi/2 - 2sin^-1 x`
`(dy)/(dx)=-2/sqrt(1-x^2)`