If $f(x,y)=f(y,x)$ then $\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(y,x)}{\partial y}$ Why is it that if $f(x,y)=f(y,x)$ then $\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(y,x)}{\partial y}$ for all $x,y$ in $\Bbb R^2$. My lecturer just went over it like it was obvious but I cant seem to come up with a proof with why it is so. I thought maybe starting from the limit definition of a partial derivative would get me somewhere then I could jumble it till they were equal but this got me nowhere.
Any help would be very much appreciated.

The formula is correct as long as you interpret it as
$$
\partial_1 f(x,y) = \partial_2 f(y,x).
$$
Indeed,
$$
\begin{align}
f(x,y) & =f(y,x) \\
f(x+h,y) &= f(y,x+h).
\end{align}
$$
Now subtract and divide by $h$.
