If `y=sin^(-1)[xsqrt(1-x)-sqrt(x)sqrt(1-x^2])` and `0

Explanation

`y=sin^(-1)[x sqrt(1-x)-sqrt(x)sqrt(1-x^(2))]," where "0lt x lt 1`
`=sin^(-1)[xsqrt(1-(sqrt(x))^(2))-sqrt(x)sqrt(1-x^(2))]`
`=sin^(-1)x-sin^(-1)sqrt(x)`
`[because sin^(-1) x-sin^(-1)y=sin^(-1)(xsqrt(1-y^(2))-ysqrt(1-x^(2)))]`
Differentiating w.r.t.x, we get
`(dy)/(dx)=(1)/(sqrt(1-x^(2)))-(1)/(sqrt(1-(sqrt(x))^(2)))(d)/(dx)(sqrt(x))`
`=(1)/(sqrt(1-x^(2)))-(1)/(sqrt(1-x))xx(1)/(2sqrt(x))`