# Are two definitions of differential function equivalent?

Definition 1:

A function $f(x, y)$ is differentiable at the point $(x_0,y_0)$ if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ exist and $\Delta f = f(x_0 +\Delta x, y_0 +\Delta y) − f(x_0,y_0)$ can be written in the form $$\Delta f=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y$$ where $\epsilon_1$ and $\epsilon_2$ are functions of $\Delta x$ and $\Delta y$ such that $$\lim\limits_{(\Delta x,\Delta y)\to(0,0)}\epsilon_1=\lim\limits_{(\Delta x,\Delta y)\to(0,0)}\epsilon_2=0.$$

A function $f(x, y)$ is differentiable at the point $(x_0,y_0)$ if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ exist and $\Delta f = f(x_0 +\Delta x, y_0 +\Delta y) − f(x_0,y_0)$ can be written in the form $$\Delta f=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y+o(\sqrt{\Delta x^2+\Delta y^2}).$$

I wonder whether two above definitions of differential function are equivalent. Any explain are highly appreciated.