What if the key to undertanding gravity is considering the n-th derivative of position with respect to time?

There is no such thing as an instant change in velocity because that would imply infinite acceleration, the same goes with acceleration and jerk. There is almost no material covering the relationship between jerk and gravitational atraction, its effect may be negligible at large scales but if we take into account the small contributions of those tiny increments there might be something to be said about the way we understand motion.