[alecst]

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.

[fault1]

Another way to see the it is to explore it from the matrix exponential structure and the link with trig (esp odd/even functions) or example this video: https://www.youtube.com/watch?v=UWrt9Fj80Kc&list=PLlXfTHzgMR...

so much structure even in 2x2 rotations

[ajkjk]

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.