Use the Taylor formula
Thanks for your comment. Indeed, I need a parametric function in terms of x and y like what is written for (x-y)^2=x^2+y^2-2xy. However, I know there is not an exact extension for my equation. I think there is maybe a simplified form with a reasonable approximation.
Yes, an approximation is what you asked for and what FernandoRevilla gave you. Your function is f(x,y)= (x- y)^{0.5}, so its first order partial derivatives are f_x= (0.5)(x- y)^{-0.5} and f_y= -(0.5)(x- y)^{-0.5}. Evaluate those at some (x_0, y_0) and put into FernandoRevilla's formula. Of course, any approximation is not going to be equally accurate for all x and y. You need to decide where you want the approximation to be most accurate and pick x_0 and y_0 there.