@SimplyBeautifulArt: And then I noticed that you had posted this wrong answer many months ago. So many reasons why it is wrong, and it's best that you delete it.
Well it's a generalization in the sense that derivatives are treated in terms of a parameter, rather than forcing one of the coordinates to be a parameter. So sometimes curves that don't have any ordinary derivative at a point have a derivative under my framework, such as the second example.
This makes intuitive yet rigorous proofs of the various rules in the style of Leibniz actually possible. Feel free to ask anything about it and I will clarify.