A text I was reading used производная от полинома a few times. What extra detail does от add that is not present in производная полинома? Are производная от функции or сумма от чисел also acceptable, and if so how do those differ from производная функции or сумма чисел? Does this usage of от (where its presence or absence are both meaningful) also appear in non-technical speech?
In a mathematical context, there is no practical difference. Where "от" is explicitly used, it just (in my view) reinforces the genetive case. In some phrases it is customary; in some not ("сумма от чисел" is never used, it would have a different (non-sensical) meaning, see below).
You can more often see "от" when there is a dependent word, such as "производная от функции sin(x)" [hey, no MathJax here?!] (vs "производная функции" as an independent term), but this is not a specific rule.
In general usage, direct use without a preposition is roughly equivalent to english "of" (posession/belonging), whereas "от" is equivalent to "from" (origin/source). It math, this is often the same, but in many cases it is not.
First thing - "от" is not a verb, but preposition.
As for usage with derevatives - mathimatical text in Russian is very hard to read due to lots of grammatics dependences between close and distant phrases and terms. While in simple text there is no difference between using and not using "от" (it's a matter of taste), in complex text this preposition helps to avoid ambiguity.
Consider this simple example:
"не следует производить интегрирование по Жордану производной сложного полинома тригонометрических функций"
While it is still uderstandable, I (who think up this phrase!) need to read it twice. Prepositions will make it clearer for sure. In Russian such a "chain of nouns" is s stylistic error (unlike in English).
As for "сумма от чисел" - it is unacceptable, becase of plural form and dual nature of sum operation. "Сумма от числа" would be acceptable if it had sense.
Also acceptable "квадратный корень от числа" (but not used much), "интеграл от функции"
No extra details.
Some people use 'от' when they talk about algebraic or trigonometric functions (but not arithmetic though).
So 'Производная от функции', 'Интеграл от функции' are acceptable as well as 'Синус от нуля' or 'Экспонента от функции'. However, if you see it in a written text it probably means that author has poor grammar.
'Сумма от чисел', 'Произведение от чисел' aren't acceptable at all.
If you think about it semantically, производная comes from производить, to produce. So the word indicates origin or ingredients, so we use от with it, like it's "made from".
A sum though is like a collection of or a container of things. Мешок камней, сумма чисел. A sum of things.
Or for example, in Russian when we talk about square root, we say извлекать корень, to "extract a root", because извлекать (а) ИЗ (б), we tend to say квадратный корень из числа.
They're not hard and fast rules and alternative prepositions are accepted, but these are just some examples of semantic logic for those prepositions in math.
It appears in non-technical contexts. "Колесо машины" and "колесо от машины" will have the same meaning when you see a wheel lying dismantled near a car as well as on its own with no car in view. But when a wheel is mounted where it should be, you'd better say "колесо машины". The same usage is applied to almost all similar constructions. But in colloquial speech you can both use "от" or not use it without any change of meaning notwithstanding the whole and the part being together or having been laid separately.
One note to be heeded is that insofar as a kind of instrument and not the whole and its part is concerned the preposition "от" is rather used (ключ от квартиры, средство от комаров).