If $f:\Bbb R^2\to\Bbb R$, what does $$\frac{\partial^2 f}{\partial x\, \partial y}$$ mean? Does it mean $\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)$ or $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)$?
(I know that most of the time due to Schwarz theorem it is irrelevant.)
This is a matter of convention see e.g. here and here. So you should look at your lecture notes to use the definition proposed there.
In any case, if $f$ is twice continuously differentiable, then $$\dfrac{\partial}{\partial x}\!\!\left(\dfrac{\partial f}{\partial y}\right)=\dfrac{\partial}{\partial y}\!\!\left(\dfrac{\partial f}{\partial x}\right)$$