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I want to derive an expression with respect to each variable of a function $f: \mathbb R^n \rightarrow \mathbb R$. More precisely:

$\left(\prod_{i=1}^n\frac{\partial}{\partial_{x_i}}\right) g(f(x_1,x_2,\dots,x_n))$

with $n\in \mathbb N$ generic and $g: \mathbb R \rightarrow \mathbb R$ is known function (for instance $ g(y) = \arcsin(y)$).

Do you know how to get a symbolic result? I think the use of blanks __ is the right way to go, but I cannot figure out how.

Edit

I'm looking for an output result of the form (with $n$ unspecified)

$\sum_i^n h_i(\{f^{(i...)}\})$

where $i$ is a (possibly multi-)index and $h$ some function of the partial derivatives of $f$ that depends on the known function $g$.

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  • $\begingroup$ A possible duplication of mathematica.stackexchange.com/questions/7876/… $\endgroup$ – user64494 Oct 28 '19 at 12:03
  • $\begingroup$ I edited the question, Now the difference with that suspected 'duplicate' is more evident. $\endgroup$ – M. Marciani Oct 28 '19 at 13:22
  • $\begingroup$ Symbolic order derivatives were introduced as a new feature in v11.1 (look for 'D (updated)'). $\endgroup$ – Thies Heidecke Oct 28 '19 at 16:15
  • $\begingroup$ @Thies-heidecke. It looks to me that even the 'D (updated)' function doesn't do the job. Indeed, first one has to define $f(x_1,x_2,..x_n)$ specifying all variables names '$x_j$' and also the number of variables $n$ itself, only then one can apply $D$ to $f$ saying it explicitly to derive in $x_1,x_2,...$. I need a more abstract approach where you don't have to give a name to all variables of $f$ in the first step. $\endgroup$ – M. Marciani Oct 28 '19 at 17:01
  • $\begingroup$ Mathematica can't handle this unknown number of derivatives. On the other hand this is a pretty simple pure math question so it can't hurt to ask on math.stackexchange where they've probably done this already? $\endgroup$ – b3m2a1 Oct 28 '19 at 17:21
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I'm not sure if Mathematica can help you find the closed form result directly, but you can generate examples for given n as follows.

Define a generic function of an arbitrary number of variables by using a single dummy variable x

f[0][x]

the first entry 0 will keep track of derivatives and shows that no derivatives have been taken yet.

Then, define a derivative operation with a dummy variable x as follows

Dxi[input_, xi_] := D[input, x] /. Derivative[1][f[y_]][x_] :> f[y + xi][x]

this will take any input and take a derivative with respect to the dummy x, then any f[y]'[x] expressions are replaced by f[y+xi][x] which denotes that a derivative with respect to xi has been applied, e.g.,

Dxi[g[f[0][x]], x[i]]

enter image description here

With this, you can generate effective derivatives on function g[f[0][x]] with any amount of different x[i]. For example, if n=3

n=3;
gderivatives=g[f[0][x]];
Do[gderivatives=Dxi[gderivatives,x[i]];,{i,1,n}];
Collect[gderivatives,Derivative[_][g][f[0][x]],FullSimplify]

enter image description here

Staring at a few examples for lower n values you might come up with a general pattern how the derivatives arrange themselves. This will likely involve sets, subsets and permutations of labels and such.

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  • $\begingroup$ Thanks for the effort. I'm actually looking for a code that gives an output with $n$ left unspecified. But maybe you are right, Mathematica cannot handle it, as it should recognize the combinatorial pattern behind this derivative... $\endgroup$ – M. Marciani Oct 28 '19 at 20:41

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