If `y=xsin^(-1)x+sqrt(1-x^2), p rov et h a t(dy)/(dx)=sin^(-1)x`

Explanation

Let `x = sin theta => dx/(d theta) = cos theta `

Now, `y = sintheta* theta + sqrt(1-sin^2theta)`

`=> y = theta sin theta +cos theta`

`:. dy/(d theta) = theta(cos theta) +sin theta - sin theta`

`=> dy/(d theta) = theta(cos theta)`

`=> dy/dx = (dy/(d theta))/(dx/(d theta)) = (theta cos theta)/(cos theta) = theta`

As `x = sin theta`

`=> dy/dx = sin^-1x.`