Reference Request: Does there exist a definition for a "total finite difference" operator?

Question

I was just wondering if there is a total finite difference operator defined in a way that is analogous to the total derivative operator. As we know, for a function of one variable, we might define the finite difference by $$\Delta f(x_n) = f(x_{n + 1}) - f(x_n),$$ and we could conceivably treat this as a derivative operation if we consider an equally spaced grid so that $$\frac{\Delta_h f(x)}{\Delta_h\,x} = \frac{f(x + h) - f(x)}{(x + h) - x}.$$ My question is if anyone has extended this to a "Total Finite Difference" operator in the same way that we have a total derivative for $f(x,y)$ given by something like $$D_{\Delta} f(x_n,y_m) = f_{\Delta x} \Delta x_n + f_{\Delta y}\Delta y_m.$$ where $f_{\Delta x} = f(x_{n + 1}, y) - f(x_n)$ and $\Delta x_n = x_{n + 1} - x_n$ (or something like this). If this does exist, does anyone know if this type of operation plays well with summation by parts? Any help is appreciated. Thank you!