Simple multi-index question

First, I have never really liked multi-index notation.

I wish to find a neat expression for

[math]\partial_{\mathbf{A}}(f(x))^{l} = \frac{\partial^{|\mathbf{A}|}}{\partial x^{A_{1}}\cdots \partial x^{A_{n}}} (f(x))^{l}[/math].

Here [math]\mathbf{A}[/math] is a multi-index, but [math]l[/math] is an exponent.

One should be able to you the Leibniz rule recursively to get a nice expression. (But note that [math]\partial_{\mathbf{A}}[/math] is not a derivation).

Has anyone seen the answer to this anywhere? I am sure it has been calculated before now. Can anyone save me the time and effort in doing it myself? I expect it is in a book on pseudo-differential operators. (I can't get to the library today).

Thanks

It does not matter to much now, as I have found exactly what I need in paper I am reading.

Still, thanks to those people who read this post.