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Is there a name for the set of functions that make up a function's derivatives? Integrals?

For example's sake, let's say there is a fourth degree polynomial function $f(x) = 1 + x^2 + x^3 + x^4$. What would the collection $f(x)$, $f'(x)$, $f''(x)$, ... be called? Can this be extended to integration?

I tried searching for something along the lines of generalizing the set of derivatives or a generalization of gradient, but found generalizations in the multivariate or different coordinate senses, not to n'th derivatives.

Alex Firsov
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  • Apologies if the question is overly simplistic, loaded, or incorrectly tagged. I don't know enough math at this time to ask better, but would appreciate edits :) – Alex Firsov Nov 11 '20 at 23:30
  • I don't think that set of functions has a name. If you have a context in which you will need to refer to it often you can make up your own. If you [edit] the question to show us some of that context perhaps we can suggest something. – Ethan Bolker Nov 11 '20 at 23:31
  • I don't have any particular context. Just studying calculus, and figured there has to be some work on this set/space/whatever this would be called, since there's a clear pattern here for generalization. – Alex Firsov Nov 11 '20 at 23:36
  • The collection ${f(x),f^\prime(x),f^{\prime \prime} (x),\cdots}$ is called "the function $f$ and its derivatives." – mjw Nov 11 '20 at 23:38

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The answer to this question really depends on your motivation for it. If you just want a name for the sequence of pointwise values of a function and its derivatives at a point, then "the sequence of pointwise values of a function and its derivatives at a point" is an answer.

However, you probably want to do something with this sequence, thus you need a definition that hints at some more structure on it. In that case, the closest notion I can think of is the one of "Jet" ; https://en.wikipedia.org/wiki/Jet_(mathematics). It arises in the study of symmetries of differential equations, because it allows to rewrite such equations as algebraic ones; see Why is a PDE a submanifold (and not just a subset)?.

(On these things, I like the book of Peter Olver "Applications of Lie groups to differential equations").

Giuseppe Negro
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