Suppose $f(x,y)=f(y,x)$. Does it follow that $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}$?
Intuitively it seems like it must, because taking a "step" in the $x$ direction must be the same as taking one in the $y$. But when I try to prove it I get lost as to which is "really" x or y.
EDIT: I should have been more clear. What I meant was: does $\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(y,x)}{\partial y}$?
(At some intuitive level this is like doing a "find and replace" s/x/y/, but my intuition fails when taking the derivative.)