I always got the sense from physics that outside of purely mathematical constructions such as Taylor series, higher order time derivatives aren't providing much interesting information. Though I'm not sure whether this is the inherent laziness of physicist math[1] or a property of the forces in nature.
[1] since e^x = 1 + x is generally true, why'd you even need a second order derivative
I cannot remember what it's called but essentially given a target position in space the missle uses parametric data about its current position/orientation/speed and their higher derivatives to dead reckon about where it is in regards to the target.
Anyone remember what that's called? I went on a rabbit hole with it a few years ago, it's really interesting math and programming. Everything works basically stateless except for current instrument data, last position and target position from what I remember.
Kalman filters come up a lot, maybe relevant to the terms you're looking for