A nicer notion is that of the differential:
$$ \text{If} \qquad z = 5x + 3y \qquad \text{then} \qquad dz = 5\, dx + 3\,dy $$
Then if you decide to hold $y$ constant, that makes $dy = 0$, and you have $dz = 5 \, dx$.
Another notation that works well with function notation is that if we define
$$ f(x,y) = 5x + 3y$$
then $f_i$ means derivative of $f$ with respect to the $i$-th entry; that is
$$ f_1(x,y) = 5 \qquad \qquad f_2(x,y) = 3 $$
This doesn't work well with a common abuse of notation, though; sometimes people write $f(r,\theta)$ when they really mean "evaluate $f$ at the $(x,y)$ pair whose polar coordinates are $(r, \theta)$" rather than the 'correct' meaning of that expression "evaluate $f$ at $(r, \theta)$". So if you're in the habit of doing that, don't try to indicate derivatives by their position.
I confess I really dislike partial derivative notation; when one writes $\partial/\partial x$, one "secretly" means that they intend to hold $y$ constant, then when one passes it through the differential, one gets
$$ \frac{\partial z}{\partial x} = 5 \frac{\partial x}{\partial x} + 3 \frac{\partial y}{\partial x} = 5 \cdot 1 + 3 \cdot 0 = 5$$
However, the suggestive form of Leibniz notation starts becoming very misleading at this point; for example, let's compute other partial derivatives.
- $\partial z / \partial x = 5$, holding $y$ constant as the notation suggests
- $\partial x / \partial y = -3/5$, holding $z$ constant as the notation suggests
- $\partial y / \partial z = 1/3$, holding $x$ constant as the notation suggests
Then putting it together,
$$ \frac{\partial z}{\partial x} \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} = 5 \cdot \left(-\frac{3}{5}\right) \cdot \frac{1}{3} = -1 $$
This is a big surprise if you expect partial derivatives to behave similarly to fractions as their notation suggests!!!
Best Answer
$\def\part#1#2{{\partial#1\over\partial#2}}$ $\def\parts#1#2#3{{\partial^2#1\over\partial#2\,\partial#3}}$
On the left hand side of your equations, you have the symbol "$\part{\vphantom f}y\bigl(\part f x\bigr)"$. By definition this is the partial derivative of the function $\part fx$ with respect to $y$. So, upon encountering this symbol, you take the function $\part fx$ and then take its partial with respect to $y$. The natural notation of the type on the right hand side of your equations is the notation used in (2) of your post: $$\tag{3} \part{\vphantom f}y\Bigl(\part f x\Bigr)=\parts f y x. $$
I will not surmise why this is the "natural" notation, but will point out that $(3)$ gives the adopted definition for the symbol $\parts f y x$ in any calculus/analysis text, or any other "credible" source, you'll find.
I emphasise here that $(3)$ defines the symbol $\parts f y x$; that this sometimes gives an expression that equals $\parts f x y$ is irrelevant. (Of course, for certain functions, what you wrote in (1) would be correct; but its correctness would follow from the result of a theorem, not from the definition of the symbols.)