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

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`
`=>1/(sqrt(2)) lt sin theta lt 1`
`=>pi/4 lt theta lt pi/2`
`=>pi/2 lt 2theta lt pi`
`=>-pi/2 lt pi/2 -2theta lt 0`
`y=- pi/2 +2theta`
`=>y= -pi/2 +2sin^-1 x`
`(dy)/(dx)=2/sqrt(1-x^2)`