Differentiate `sin^(-1)(2xsqrt(1-x^2))` with respect to `x`
`theta in [-(pi)/(2),(pi)/(2)]` ,

Explanation

We have `y= sin^(-1) (2xsqrt(1-x^(2))`
Let `x= sin theta`,
where `theta = sin^(-1)x`, and `theta in [-(pi)/(2),(pi)/(2)]`
`therefore" " y= sin^(-1) (2 sin theta cos theta)`
` = sin^(-1) (sin 2 theta)`
`-1ltxlt-(1)/(sqrt(2))`
or`-(pi)/(2)ltthetalt-(pi)/(4)`
or`-pilt2thetalt-(pi)/(2)`
`therefore" "y=sin^(-1)( sin2theta)=sin^(-1)(-sin(pi+2theta))`
`=sin^(-1) (sin(pi-2theta))=-pi-2theta`
`therefore" "y=-x-2 sin^(-1)x`
`rArr" "(dy)/(dx)=0-(2)/(sqrt(1-x^(2)))`
=`-(2)/(sqrt(1-x^(2)))`