Differentiating matrix function in `Table` from

Question

I am trying to get an analytical expression for derivaives of an algorithm explained here. I usually use automatic diff. a lot, but since this part of function will not change, I thought to implement analytical solution if possible. Below is my attempt:

Mf[Dm_] := Table[(Dm[[i, 1]] + Dm[[1, j]] - Dm[[i, j]])/2, {i, 
   Dimensions[Dm][[1]]}, {j, Dimensions[Dm][[1]]}]
func[Mm_] := Eigenvectors[Mm] . DiagonalMatrix[Sqrt[Eigenvalues[Mm]]]
D[func[Mf[Dm]], Array[Dm, {n, n}]]

I get the following errors

How to do it right? I would ideally like an analytical expression giving derivatives w.r.t Matrix elements of Dm.
Is it possible to use mathematica for analytical expressions of ML neural nets? I had an idea to train NN using pytorch, then using final trained model to get analytical derivatives from mathematica and using those for more efficient production environment (basically switching off autograd as we have a definite Linear algebraic form of expression).

I am using Mathematica first time for some actual problem so sorry if it is a naive question.

Should you have {j, Dimensions[Dm][[2]]} instead of the first part? — IntroductionToProbability, Nov 16 '22 at 16:30
Ideally yes. But since Dm will always be a square matrix so I thought this might simplify it. Only implementing equal dimension restriction. — ipcamit, Nov 16 '22 at 16:36
this might be helpful https://mathematica.stackexchange.com/questions/138708/differentiating-functions-of-vectors-matrices — IntroductionToProbability, Nov 16 '22 at 16:39
That requires a package called MatrixD, I will look into it. But does this mean we can differentiate through Table command? How does one write Matrix indices based functions then? — ipcamit, Nov 16 '22 at 18:02

Differentiating matrix function in `Table` from

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