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 :(

If $f$ is defined by $f(x,y)$, the operation $\frac{\partial f}{\partial x}$ means that you differentiate $f$ with respect to $x$. With your example this will become $2xy$. Now, $\frac{\partial f}{\partial x}(y,x)$ means, that you FIRST differentiate with respect to $x$ (which was $2xy$) and AFTERWARDS plug in $(x,y)=(y,x)$. Therefore we obtain $\frac{\partial f}{\partial x}(y,x)=2yx(=2xy)$.
Now, if $g(x,y)=f(y,x)$, then $g(x,y)=y^2x$. Again, $\frac{\partial g}{\partial x}(x,y)$ means that you have to differentiate $g$ with respect to $x$ and afterwards plug in $(x,y)=(x,y)$ (so this won't change anything). Thus, we obtain $\frac{\partial g}{\partial x}(x,y)=y^2$.
