`cos^(-1)(2xsqrt(1-x^(2)))`

Explanation

Let `y=cos ^(-1)(2xsqrt(1-x^(2)))`
Put ` x=sin theta`
`thereforetheta=sin^(-1)x `
`y=cos^(-1)(2sinthetasqrt(1-sin^(2)theta))`
`y=cos^(-1)(2sin thetacostheta)`
`y=cos^(-1)(sin 2theta)`
`y=cos ^(-1)[cos ((pi)/(2)-2theta)]`
`y=pi/2-2theta`
`y=pi/2-2sin ^(-1)x`
`therefore (dy)/(dx)=(-2)/(sqrt(1-x^(2)))`