Take the factorial function defined in one of the usual ways (recursive or iterative). How do you take the derivative of it, using this paper? Do you magically

Take the factorial function defined in one of the usual ways (recursive or iterative). How do you take the derivative of it, using this paper? Do you magically get the gamma function this way, too? I just skimmed the paper without getting close to answering these questions -- anyone who does already understand, maybe explaining this concrete example could help the rest of us? Thinking about the problem in my naive way earlier this year, I noticed that the usual if-then-else expression is a special case of linear interpolation, at least when you're dealing with numbers. ("lerp by t between a and b" generalizes "if t then b else a", taking false=0 and true=1.) But it's not so obvious how to handle recursion and nonnumeric data: I guess there's a hint in that the usual theory of programs has Scott continuity (https://en.wikipedia.org/wiki/Continuous_function https://en.wikipedia.org/wiki/Continuous_function) while differentiable functions are continuous in the reals, but I'm mathematically undereducated.