I've taken multivariate calculus and am wondering if I can see a specific function where the order of taking the partial derivative matters. I've been told that there are some exceptions where $ \dfrac{\partial ^2 f}{\partial x \partial y} \ne \dfrac{\partial ^2 f}{\partial y \partial x} $, so I'm curious to see what this looks like.
EDIT: And why would this true?
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$\begingroup$Consider the function $f(x,y):=\frac{xy^3}{x^2+y^2}$, with the additional constraint that $f(0,0):=(0,0)$ -- you can check that this function is continuous and differentiable for all $(x,y)$, from the quotient rule we have $$\frac{\partial f}{\partial y}(x,y)=\frac{3xy^2}{x^2+y^2}-\frac{2xy^4}{(x^2+y^2)^2},$$ and at $y=0$ we have $\frac{\partial f}{\partial y}(x,0)=0,$ thus $$\frac{\partial^2f}{\partial x\partial y}(0,0)=0.\tag1$$ On the other hand, by the quotient rule again $$\frac{\partial f}{\partial x}(x,y)=\frac{y^3}{x^2+y^2}-\frac{2x^2y^3}{(x^2+y^2)^2},$$ and hence $\frac{\partial f}{\partial x}(0,y)=y$. Thus $$\frac{\partial^2 f}{\partial y\partial x}(0,0)=1.\tag2$$ So you can see from $(1)$ and $(2)$ that $\frac{\partial ^2 f}{\partial x \partial y} \neq \frac{\partial ^2 f}{\partial y \partial x}$.
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