Differentiate `sin^(-1)(2xsqrt(1-x^2))`,`-1/(sqrt(2))ltxlt1/(sqrt(2))` with respect to `x` :

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)`
`-(1)/(sqrt(2))ltxlt(1)/(sqrt(2))`
or`-(pi)/(4)ltthetalt(pi)/(4)`
or` -(pi)/(2)lt2thetalt(pi)/(2)`
`therefore" "y=sin^(-1) (sin2theta)`
`=2theta`
`=2 sin^(-1)x`
`rArr" "(dy)/(dx)=(2)/(sqrt(1-x^(2)))`