How to prove that $f(x,y)=xy$ is differentiable at $(x_0,y_0)$? 
Prove that $f(x,y)=xy$ is differentiable at $(x_0,y_0)$ using the $\epsilon$ definition.

We can use the definition of differentiability: 
$$
f(x_0+\Delta x, y_0+\Delta y)-f(x_0,y_0)=f_x(x_0,y_0)\cdot\Delta x + f_y(x_0,y_0)\cdot \Delta y+\epsilon_1\cdot\Delta x+\epsilon_2\cdot\Delta y
$$
According to the above formula we get:
$$
x(y-y_0)-x_0(y-y_0)=(x-x_0)(y-y_0)=\epsilon_1\cdot(x-x_0)+\epsilon_2\cdot(y-y_0)
$$
By comparing the coefficients we have that $\epsilon_2=x-x_0$ thus:
$$
\lim_{(x,y)\to(x_0,y_0)} (x-x_0)=0
$$
But what about $\epsilon_1$?
We can conclude that $\epsilon_1(x-x_0)=0$ but we can't know if $\epsilon_1$ necessarily goes to $0$.
 A: $f$ is differentiable at $(x_0,y_0)$ if the difference $\,(x-x_0)(y-y_0)\,$ between $f(x,y)-f(x_0,y_0)$ and its linear approximation $y_0(x-x_0)+x_0(y-y_0)$ is $o\bigl(\lVert(x-x_0,y-y_0)\rVert\bigr)$.
To see this, set $x-x_0=r\cos\theta$, $y-y_0=r\sin\theta$ $\;(r>0,\;0\le \theta<2\pi)$. This difference is $(x-x_0)(y-y_0)$, so we have
$$\frac{\lvert(x-x_0)(y-y_0)\rvert}{\lVert(x-x_0,y-y_0)\rVert} =\frac{r^2\lvert\cos\theta\sin\theta\rvert}r=r\lvert\cos\theta\sin\theta\rvert\le r\to 0.$$
A: We have that $\nabla{f}(x_0,y_0)=(y_0,x_0)$
We'll use the definition of differentiability of a function at a particular point.
So from the definition:  $$\frac{|f((x_0,y_0)+(h_1,h_2))-f(x_0,y_0)-<\nabla{f}(x_0,y_0),(h_1,h_2)>|}{\sqrt{h_1^2+h_2^2}}$$ $$=\frac{|(x_0+h_1)(y_0+h_2)-x_0y_0-y_0h_1-x_0h_2|}{\sqrt{h_1^2+h_2^2}}$$ $$=\frac{|x_0y_0+x_0h_2+y_0h_1+h_1h_2-x_0y_0-y_0h_1-x_0h_2|}{\sqrt{h_1^2+h_2^2}}$$ $$=\frac{|h_1h_2|}{\sqrt{h_1^2+h_2^2}}$$
Now use the fact that $|h_1h_2| \leqslant \frac{1}{2}(h_1^2+h_2^2)$
and take $\delta=2 \epsilon$ and you are done.
Also $<.,.>$ denotes the usual inner product in $\mathbb{R}^2$.
