A perhaps nicer way to look at things[0] is to hold onto your base points explicitly and say Df:: a -> (b, a -o b) = (f(p),A(p)) where f(p+v)≈f(p)+A(p)v. Then you retain the information you need to define composition Dg∘Df=D(g∘f)=(Dg._1∘Df._1, Dg(Df._1)_.2∘Df._2). i.e the chain rule.

[0] which I learned from this talk https://youtube.com/watch?v=17gfCTnw6uE

It's deplorable that we can't write in Latex or something similar here in 2025, and have to resort to the gobbledygook above.

Yes! I love Conal Eliot’s work. The one you wrote is the compositional derivative which augments the regular derivative by also returning the function itself (otherwise composition won’t work well). For anyone interested look up “the simple essence of automatic differentiation”.

I respect the time you spent to write such a post with all those limited input alternatives (bowes).