If you can take an infinite number of derivatives of position, then why is it so hard to visualize/think of real world examples beyond the jerk? Does the function essentially become meaningless to the physical world?

Question

If you can take an infinite number of derivatives of position, then why is it so hard to visualize/think of real world examples beyond the jerk? Does the function essentially become meaningless to the physical world?

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Answer 1

Higher order derivatives basically give us better and better approximations to things, but you usually only need a few to get a pretty decent approximation and so that is what happens.

However, there is a sense in which higher order derivatives yield less and less information. You can tell by trying to make sense of an “infinite order derivative”. That is, what would it mean to take infinitely many derivatives and what information do you get from them? If you know the answer to this question, then you can, in a sense, recognize higher and higher order derivatives as getting closer and closer to that infinite-order information.

Well we can, in a way, make sense of the notion of an “infinite order derivative” (see [here](https://math.stackexchange.com/questions/362873/is-there-any-meaning-to-an-infinite-derivative)). The information you get from it, however, is basically nothing because, for any function f(x) (such that the following values are defined) the “infinite order derivative” will always look like f^([∞])(x) = A*e^(x) for some number A. This number A is the limit of the derivatives f^([k])(0) as k goes to infinity. This means that the infinite order derivative smooths out the entire function, basically forgetting all information except that one value. This means that, if you take too many derivatives then eventually you are not getting any new information and are merely zeroing in on this single remaining value. It can be hard to know when this “eventually” is, but it generally does mean that the first few derivatives contain a reasonable chunk of information about a function and so you don’t need to go beyond that.

So, in a way, we *don’t* need more than a few derivatives in physics to understand mostly everything about what is going on – even beyond just the simple approximation results.