Let's say I have a function $f\left(x_{1},\dots,x_{n}\right)$ which takes $n$ arguments. Now, I want to take the partial derivative: $$ \frac{\partial}{\partial x_{3}}\frac{\partial}{\partial x_{7}}\frac{\partial}{\partial x_{9}}\frac{\partial}{\partial x_{12}}\frac{\partial}{\partial x_{14}}\frac{\partial}{\partial x_{16}}f$$ for example (so it is a mixed partial derivative with respect to arguments 3,7,9,12,14, and 16). In this case I can write it and it isn't so bad. But I am looking for a more compact notation for when I have the indices in a set $I=\left\{ 3,7,9,12,14,16\right\} $.
I am looking for something similar to when we take an iterated product or sum and we use $\prod_{i\in I}$ or $\sum_{i\in I}$, so maybe something like $\partial_{i\in I}$? but I am wondering if this a standard notation or whether there is some standard notation for this?
Another case in which I want to use such a notation for a repreated operator is with finite differences. So say I want to take the finite difference with respect to some arguments of the function, and want to do it iteratively for some set of indices.
EDIT: To clarify, I am looking for a way to do this for an arbitrary set $I$, which could be much larger than 6 elements.
You have to be aware that these operators do not necessarily commute (there are conditions on $f$ though that make them commute).
In the face of non-commutativity you can't use an index set as these are not ordered and the order of the factors are important.
Apart from that I haven't seen the use of $\prod$ and $\sum$ in rings, but I suppose there should be little room for misunderstanding. If that's the case the reasonable way to write it would be:
$$\left(\prod_{j=3,7,9,12,14,16}\partial_j\right)f$$
note the parenthesis - you don't want to confuse it with $(\partial_3 f)(\partial_7 f)(\partial_9 f)\cdots$
However if you're looking for a compact notation you also can use:
$$\partial_3\partial_7\partial_9\partial_{12}\partial_{14}f$$
which to me looks a bit more compact (it relies on that the operator itself has a fairly compact notation). You can also in this case use $$f'''''_{3,7,9,12,14}$$
which is even more compact (but it is special for the partial derivative operator).