Is there a formal term for the class of operators that behave similarly to the derivative?

I know of several different "types" of differentiation. There is the "standard" derivative conventionally used in calculus. There is the product derivative which is the operator going hand in hand with the product integral. There is also the class of semiderivatives which (if I understand right) are any linear interpolation of the left and right hand derivatives. There is also the fractional order derivatives, of which there are several(?) variations. I feel like the list could go on and on, and if it doesn't then fair enough.

Regardless these all have a theme of being "derivatives". However, they aren't technically derivatives of functions. Some are other operators altogether. Is there a specific formal name to this family of operators and what determines whether or not they fall into the set. Is it human choice or is there a generic set of rules it has to follow? I'm generally intrigued by these types of operators, but I'm curious as to what general rule or theory or framework makes them all in some way "equivalent". If someone could please point me in the right direction that would be greatly appreciated.

• "Differential operators"? "Derivations"? I'm not entirely sure that I understand the question... – Xander Henderson Oct 7 '17 at 3:06
• @XanderHenderson Both of those seem like possible answers although I suspect they are both the same thing under different names? What specifically do you not understand about the question? I can't edit and/or clarify if you don't say what is confusing you. :-) – The Great Duck Oct 7 '17 at 6:45
• If you can't or won't define rigorously any of the notions you are mentioning, you should tag that "soft question". – Professor Vector Oct 7 '17 at 8:52
• Your first title was way better, and your question is perfectly reasonable. I don't think there is any formal name for this family of operators you're talking about, I think people just use the term "derivative" in many contexts. – Joppy Oct 8 '17 at 0:01
• @Joppy you're welcome to take up the issue of it being bad title with the other guy. At this point I just accept all suggestions because it seems that whenever I don't people get angry. Easier to keep the peace. – The Great Duck Oct 8 '17 at 4:25