Differentiate `sin^(-1)sqrt(1-x^2)` with respect to `cos^(-1)x ,` if (i)`x in (0,1)` (ii) `x in (-1,0)`

Explanation

Given `x in(-1,0)`


However, `x=cos theta`.


`Rightarrow cos theta in(-1,0)` `Rightarrow theta in(frac{pi}{2}, pi)` Hence, `u=sin ^{-1}(sin theta)=pi-theta .` `Rightarrow u=pi-cos ^{-1} x`


On differentiating `u` with respect to `x`, we get



`frac{d u}{d x}=frac{d}{d x}(pi-cos ^{-1} x)`


`Rightarrow frac{d u}{d x}=frac{d}{d x}(pi)-frac{d}{d x}(cos ^{-1} x)`



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