Finding rules for two functions

Finding rules for two functions

What are rules for $\alpha(\Delta x, \Delta y) , \beta(\Delta x, \Delta
y)$ functions so that $f(x,y)=\mathrm{e}^{xy}$ to be differentiable on
$\mathbb{R}^2$ by the following definition?
$\textbf{DEFINITION}$: Suppose that for every point $(a+\Delta x, b+\Delta
y)$ in a deleted neighborhood of $(a,b)$ we can write: $f(a+\Delta x,
b+\Delta y)-f(a,b)=A\Delta x+\Delta y +\alpha(\Delta x, \Delta y)\Delta
x+\beta(\Delta x, \Delta y)\Delta y$ where $A,B$ are constant, and
$\alpha(\Delta x, \Delta y) , \beta(\Delta x, \Delta y)$ are functions of
$\Delta x, \Delta y$ such that $\displaystyle\lim_{(\Delta x, \Delta y)\to
(0,0)}\alpha(\Delta x, \Delta y) =\lim_{(\Delta x, \Delta y)\to
(0,0)}\beta(\Delta x, \Delta y) =0 $, then $f$ is said to be
$\underline{differentiable}$ at $(a,b)$.
$\textbf{Remark}$: We should be find $A,B$ , and functions $\alpha(\Delta
x, \Delta y) , \beta(\Delta x, \Delta y)$ at any point $(a, b)$ . Of
course I know that $A=\dfrac{\partial f}{\partial x}(a,b) \ , \
B=\dfrac{\partial f}{\partial y}(a,b)$ . But What are $\alpha(\Delta x,
\Delta y) , \beta(\Delta x, \Delta y)$ functions, exactly?