Backward propagation in non analytical functions

Hello everyone.
I’m facing the following problem I don’t really know how to solve.
I have a function that is defined as: E_{pred} = min(eigenvalue(V)), where V is an NxN matrix.
One of the elements of V, say V_{ij} is a feed-forward NN which I am trying to train.
I use E_{pred} to calculate the MSE and numerically I can get to the derivative d(MSE)/d(V_{ij}).

My problem now is. Where should I “load” this derivative to perform the backward propagation?

Thank you very much.