Ok thank you. My numerical estimate is way different from the analytical counter-part, and more likely to be wrong. I was expecting the numerical estimate to be closer to the correct answer. The issue I am having is either a linear approximation is not good enough, or the quantities are wrong then.

It sounds like you agree that the derivative tensor components can indeed be extended from a simple y = f(x) case since it is just an array of functions?

In any case, my question is mainly whether if it can be extended to a tensor like that

@CarloBeenakker you are correct. What I was thinking is if my step size is small, every point in the derivative can be approximated as linear, so I am simply calculating a linear slope with that.