[constantcrying]

>Is this type of analysis a part of a particular mathematical heritage ?

It is a mixture of two very much related areas of mathematics. Analysis, called calculus in the US, and numerics.

The ideas behind automatic differentiation arise from the question of how to compute the derivative of a function on a computer. The "derivative" part is the Analysis part and the "on a computer" part is the numerics.

As it turns out writing down the formal definition of the derivative and approximating it on a computer has many undesirable properties. So alternative approaches, like AD, were developed. But AD is much older than the recent Neural network trend.