It would let you represent physical systems as normal Haskell functions and get accurate derivatives "for free". At least in the one-dimensional case it's probably simpler than numeric differentiation while also letting you worry far less about accuracy and stability.
I've never worked this through to a full conclusion, but you could even write it in a way that would let you get symbolic differentiation out of it too.
Yes, if you get to automatic differentiation by overloading your operators to also take a “differential” type, you can further overload them to do symbolic arithmetic and then symbolic differentiation falls out for free.
Makes sense. I initially thought, that automatic differentiation would be mostly useful for long derivative chains, but of course even for single derivatives it does have advantages.
Why?