Find `(dy)/(dx),` if `y=sin^(-1)[xsqrt(\ 1-x)-\ sqrt(x)\ sqrt(1-x^2)\ ]\ `

Explanation

`y = sin^-1(xsqrt(1-x)+sqrtxsqrt(1-x^2))->(1)`

Let `x = sin theta`

Then, `cos theta = sqrt(1-x^2)`

Let `sqrt x = sin phi`

then, `cos phi = sqrt(1-x)`

Putting these values in (1),

`=>y = sin^-1(sinthetacosphi + costheta sinphi)`

`=>y = sin^-1(sin(theta+phi))`

...