Mixed partial derivatives with respect to n variables

Question

I have the following generic set of variables:

x = {x1, x2, ..., xn}
nu = {0, 1, 2, ..., n - 1}
order(nu) := Total @ nu;

I would like to calculate:

$$ \frac{\partial^{\operatorname{order}(v)}}{\partial_1^{\nu_1} \partial_2^{\nu_2} \dots \partial_n^{\nu_n}} $$

With a generic $x$. Indeed, I would like a function of this sort:

MyMixedDerivative[f,x_?VectorQ]:= ...

Suggestions?

Answer 1

OK. If one looks at the documentation of Derivative, it should be found that Derivative accepts two arguments. The first is a Sequence of orders of derivatives to be taken and the second is a Function or a Symbol representing a function, say f. So when the order argument is passed to Derivative, it should be destructured from a List to a Sequence:

testx = {x1, x2, x3}; testf[x1_, x2_, x3_] := x1^3 x2^2 x3^2;
Derivative[Sequence @@ Table[1, {i, 1, 3}]][testf][Sequence @@ testx]

which gives

12 x1^2 x2 x3

Moreover, if one digs deeper in the "Details" part of the documentation, s/he should find out that the above description of requirement is not quite accurate. The true requirement is to achieve a structure consistency between the order argument and the argument of f. So below codes work as well

testf2[{x1_, x2_, x3_}] := x1^3 x2^2 x3^2;
Derivative[Table[1, {i, 1, 3}]][testf2][testx]

testf3[{x1_, x2_}, x3_] := x1^3 x2^2 x3^2;
Derivative[{1, 1}, 1][testf3][{x1, x2}, x3]

and other combinations of Sequence and List as the argument.