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

Given that,

`y=(xsin^(-1)x)/(sqrt(1-x^2))+logsqrt(1-x^2)`

`y=(xsin^(-1)x)/(sqrt(1-x^2))+frac{1}{2}log(1-x^2)`

Differentiating w.r.t. `x`

`frac{dy}{dx}=frac{sqrt(1-x^2)frac{d}{dx}(xsin^(-1)x)-(xsin^(-1)x)frac{d}{dx}(sqrt(1-x^2))}{1-x^2}-frac{x}{1-x^2}`

`frac{dy}{dx}=frac{sqrt(1-x^2)(sin^(-1)x+frac{x}{sqrt(1-x^2)})-(xsin^(-1)x) frac{(-2x)}{2sqrt(1-x^2)}}{1-x^2}-frac{x}{1-x^2}`

`frac{dy}{dx}=frac{sqrt(1-x^2) sin^(-1)x+x+frac{(x^2sin^(-1)x)}{sqrt(1-x^2)}}{1-x^2}-frac{x}{1-x^2}`

`frac{dy}{dx}=frac{(1-x^2) sin^(-1)x + x sqrt(1-x^2)+(x^2sin^(-1)x)}{(1-x^2)^(3/2)}-frac{x}{1-x^2}`

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