Well! I don't have a PhD at all, but I think the exponential map is a surprisingly intuitive operation once you get used to it. It works on all kinds of operators. I'd argue for teaching it much earlier, in intro calculus, to introduce the idea that e^(a d_x) f(x) = f(x + a). But it still feels like there is some magic in the fact that works that I don't have a really satisfying explanation for (although it is easy to see by expanding the Taylor series).
The magic comes from the fact that you can decompose a translation (as in your example) into a bunch of little ones. So you want an operator that has the property that F(a)g(x) = g(x+a) = F(a/N)^N g(x). Equating F(a) to F(a/N)^N (for any N) reveals the exponential structure. I’m sure there are other ways but this is the first that comes to mind. You can also try using a very small translation F(da) and that will give you some insight too.
Yeah, I know how to derive it, but it still feels very unsatisfying to say: voila, you can put derivatives inside functions. It would be a hard sell to an intro calculus student, even though the concept would be very useful at that level.
