If `u=sin^(-1)((2x)/(1+x^2))` and `v=tan^(-1)((2x)/(1-x^2))` , where `-1 < x < 1` , then write the value of `(d u)/(d v)` .

Explanation

`text { We have, } {u}=sin ^{-1}(frac{2 x}{1+x^{2}}) text { and } v=tan ^{-1}(frac{2 x}{1-x^{2}}) `

`Rightarrow frac{d u}{d x}=frac{2}{1+x^{2}} text { and } frac{d v}{d x}=frac{2}{1+x^{2}} `

`therefore frac{d u}{d v}=frac{frac{d u}{d x}}{frac{d v}{d x}}=frac{2}{1+x^{2}} times frac{1+x^{2}}{2}=1`