Partial derivative notation I still have a little problem with notation for partial derivatives.
Let
$$
f(x,y) = x^2y
$$
What do you think that this should equal to?
$$
\frac{\partial f}{\partial x}(y,x) =\, ?
$$
There are two options $2yx$ or $y^2$.
Do you think that following is the same?
$$
\frac{\partial f(y,x)}{\partial x}= \,?
$$
And now take substitution $g(x,y)=f(y,x)$. What is following?
$$
\frac{\partial g}{\partial x}(x,y)=\,?
$$
I would love to hear your opinions?

Based on DanielV comment, I need answer for things like these
$$
\frac{\partial f(y,z)}{\partial z}
$$
$$
\frac{\partial f(f(y,z),z)}{\partial z}
$$

I get constantly confused during physics lectures because of this :(

Strictly going by the rule that in $\frac{\partial f}{\partial *}(*,\cdots,*)$, you always perform the derivative operation first and then worry about evaluation, the answer must be $2xy$.  This rule must be followed, otherwise, expressions like $\frac{\partial f}{\partial y}(17)$ don't make any sense.  The Leibnitzian notation is an unfortunate one to begin with and its extension to partial derivatives is bordering on nonsense.  The Eulerian notation really shows its virtues in these cases.  For the example above, we would write $D_1 f(y,x)$, which is completely unambiguous. 
