Differentiate `sin^(-1)(2xsqrt(1-x^2))` with respect to `x` if `-1/(sqrt(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)`
(i) `-(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)))`
(ii) `(1)/(sqrt(2))ltxlt1 or(pi)/(4)ltthetalt(pi)/(2)or(pi)/(2)lt2thetaltpi`
`therefore" "y=sin^(-1)(sin 2theta)=(sin^(-1))(sin(pi-20))=pi-2theta`
`therefore" "y=pi-2 sin^(-1)x`
`rArr(dy)/(dx)=0-(2)/(sqrt(1-x^(2)))=-(2)/(sqrt(1-x^(2)))`
(iii) `-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)))`