Maybe you missed my point but I was talking about calculus in the theoretical sense i.e study of functions in the reals.
In every way you are referring to people using calculus in an applied sense always been reliant on linear algebra.
Python libraries for doing backpropagation is built on numerical techniques that computes and is definitely not just manipulating symbols and calling it chain rule.
No, back propagation is just the chain rule from calculus. d(f(g)) = df(g)dg. It is applied to linear algebra, but with linear algebra alone you will never arrive to it.
You can just admit you don't understand calculus (and go and learn it if you care). No reason to go around doing mental gymnastics about it.
In every way you are referring to people using calculus in an applied sense always been reliant on linear algebra. Python libraries for doing backpropagation is built on numerical techniques that computes and is definitely not just manipulating symbols and calling it chain rule.