Which approximation expression was more preciece? Consider the expression
$$\lim_{h\rightarrow 0} \frac{f(x+h, y+h) -f(x,y)}{h} $$
Does part of the numerator better approximated as
$$f(x+h, y+h)\approx f(x,y)+h\partial_xf(x,y)+h\partial_yf(x,y)$$
or
$$f(x+h, y+h)\approx f(x,y)+h\partial_xf(x,y+h)+h\partial_yf(x+h,y)$$
They seemed to be equivalent at the first order on the paper if further expanding to $f(x,y)$, but the numerical expression might be a bit different.(For example, the Runge–Kutta methods did perform better some cases.)
Would the second approximation "formally" be more precise?
(A reference of the Taylor expansion for multivariable around the $f(\vec a)$)
The question came up because one wanted to figure out the value of a "derivative" where two argument was coupled
$$\frac{f(x,y)}{\partial x+y}$$
where an additional condition was imposed on $x$ and $y$ with $x<<y$.

$f_0=f(x+h, y+h)$, $f_1=f(x,y)+h\partial_xf(x,y)+h\partial_yf(x,y)$, $f_2=f(x,y)+h\partial_xf(x,y+h)+h\partial_yf(x+h,y)$.
$f_0=f_1+\frac{h^2}{2} \partial^2_{xx} f(x,y)+\frac{h^2}{2} \partial^2_{yy} f(x,y)+h^2 \partial^2_{xy} f(x,y)+o(h^2)$, $f_2=f_1+2 h^2 \partial^2_{xy} f(x,y)+o(h^2)$.
Precision of $f_1$ and $f_2$ depends on function itself. Suppose $f(x,y)=(x-y)^2$, then $\partial^2_{xy} f(x,y)=-2$, $\partial^2_{xx} f(x,y)=\partial^2_{yy} f(x,y)=2$, then $f_0=f_1+o(h^2)$, $f_2=f_1-4h^2+o(h^2)$.
