If `y=(xsin^(-1)x)/(sqrt(1)-x^2)+logsqrt(1-x^2)` , then prove that `(dy)/(dx)=(sin^(-1)x)/((1-x^2)^(3/2))`

Explanation

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

Now, `y = (theta sin theta)/sqrt(1-sin^2theta) + log sqrt(1-sin^2theta)`

`=>y = (theta sin theta)/cos theta +log cos theta`

`=>y = theta tan theta +log cos theta`

`=>dy/(d theta) = tan theta + theta(sec^2 theta)+1/costheta(-sintheta)`

`=>dy/(d theta) = tan theta + theta/(cos^2theta)-tan theta`

`=>dy/(d theta) = theta/(cos^2theta)`

`=> dy/dx = (dy/(d theta))/(dx/(d theta)) = (theta/(cos^2theta))/(cos theta)`

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