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.