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If y=sin^(-1)[xsqrt(1-x)-sqrt(x)sqrt(1-x...

If `y=sin^(-1)[xsqrt(1-x)-sqrt(x)sqrt(1-x^2])` and `0 < x < 1,` then find `(dy)/(dx)`

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`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))`
`=>y = theta+phi`
`=>y = sin^-1x+sin^-1sqrtx`
Now, differentiating both sides w.r.t. x,
`=>dy/dx = 1/sqrt(1-x^2)+1/sqrt(1-x)**1/(2sqrtx)`
`=>dy/dx = 1/sqrt(1-x^2)+1/(2sqrt(x(1-x))`
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