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If y=cos^(-1){x\ sqrt(1-x)+sqrt(x)sqrt(1...

If `y=cos^(-1){x\ sqrt(1-x)+sqrt(x)sqrt(1-x^2)}` and `0 < x < 1`. Find `dy/dx` .

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Given that,
`y=cos^(-1){x\ sqrt(1-x)+sqrt(x)sqrt(1-x^2)}`
Let `x = cos theta`
Then, `sin theta = sqrt(1-x^2)`
Let `sqrt x = cos phi`
then, `sin phi = sqrt(1-x)`
Putting these values in (1),
`=>y = cos^-1(costhetasinphi + sintheta cosphi)`
`=>y = cos^-1(sin(theta+phi))`
`=>y=cos^-1[cos(pi/2-(theta+phi))]`
`=>y = pi/2-(theta+phi)`
`=>y = pi/2-cos^-1x-cos^-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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