# What do you call a function differentiated with respect to all of its arguments?

Just a simple question.

Let $f(x_1, x_2, \ldots, x_n)$ be a smooth function. Is there a particular name for the function

$$\frac{\partial^n f}{\partial x_1 \, \partial x_2 \cdots \partial x_n}$$

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I don't know any particular name for this. But if you want to think about $n$th derivatives of products/compositions/quotients/whatever, then _first_ think about the problem with _this_ kind of $n$th derivative, and afterwards with $\partial^n f/\partial x^n$, etc. –  Michael Hardy Oct 28 '12 at 3:09
It doesn't usually have a name. Since it's not coordinate-change invariant (see the chain rule), it's not a geometrically-interesting concept. –  Ryan Budney Oct 28 '12 at 3:17

This does not have a name. We call it the $n$-th partial difference of $f$ w.r.t. the vector $x$ or variables $x_1$, $x_2$, ..., $x_n$.