TITLE: Better Notation for Partial Derivatives
QUESTION [10 upvotes]: I'm constantly seeing questions here where people are confused about the notation $\frac {\partial f}{\partial x}$ or $\frac {\partial f}{\partial x} (x,y)$ or $\frac {\partial f(x,y)}{\partial x}$.  Is there some better notation which exists for partial derivatives?  If not, can anyone suggest one?
Problems with this notation:

Is the derivative evaluated at the point $(x,y)$ or is that part of the definition of the function?  If it's evaluated at the point, then should we really use $x$ as the name of the first parameter of the function?  If it's just a part of the definition of the function, how do we specify where the function is evaluated?  And if we leave off the $(x,y)$, how do we even know that $x$ is supposed to be the first parameter as opposed to anything else?
The same problem that $\frac {df}{dx}$ shares, it looks like division.  But of course $\frac {\partial f}{\partial x}$ is not as simple as division: it's a shorthand for $\lim_{h\to 0} \frac {f(x+h,y)-f(x,y)}{h}$.  This leads students to think that $\frac {\partial f}{\partial g}\frac {\partial g}{\partial x} = \frac {\partial f}{\partial x}$ and other such "cancellations" hold.  However, an operator which looks like division isn't entirely a bad thing, in that you get the correct units when doing a physical problem -- obviously this is because the definition does include a division, but only as part of a limit.
Kind of a nitpick, but $\frac {\partial^n f}{\partial x^n}$ looks awfully weird to me.  And worse, it could be confusing to some who see the "denominator" as $\partial(x^2)$ instead of $(\partial x)^2$.

Other notations include $f_x$, $\partial_x f$, and $D_x f$.  I don't like $f_x$ because that looks like an indexed function to me, not like $f$ is being operated on by a differential operator.  $\partial_x f$ and $D_x f$ both look like operators acting on $f$ but don't tell you the units and we get back to "can we really use $x$ to refer to the first parameter of $f$ outside of the definition of $f$"?
Does anyone know of a better notation?  Can anyone come up with one?

REPLY [6 votes]: In case of doubt use  $(\partial_{1}f)(x,y)$, a function of $(x,y)$ whose value at $(a,b)$ is  $(\partial_{1}f)(a,b)$.
  The penalty for not doing so is having your readers scratch their heads   over$$\frac {\partial  f}{\partial x } (y,x) \quad ??$$   You can then iterate:  define $\partial_{12}f=\partial_{1}(\partial_{2}f)$, etc.
A pleasant result is that for a twice continuously differentiable function we have $\partial_{12}f=\partial_{21}f$ so that the convention underlying the order in which we take partial derivatives doesn't matter: some authors define $\partial_{12}f=\partial_{2}(\partial_{1}f)$ .