Why are they called Rules of Differentiation and not Rules of Derivation Regarding Differentiation, Derivation, and Integration. I think semantics are SUPER important and I'm confused on the precise scope of these three words. Is derivation a type differentiation, or visa versa? Is integration considered a type differentiation or is it something completely different? The following is my understanding so far.

The non-productive suffix -ation is used to form nouns meaning "the action of (a verb)" or "the result of (a verb)".

I used multiplication as a control.




Verb
Adjective
Noun w/ non-productive suffix




Multiply
Multiple
Multiplication


Differentiate
Differential
Differentiation


Derive
Derivative
Derivation


Integral
Integrate
Integration




My main problem is this.
My textbook gives me The Rules of Integration to find the Integral of a function.
So why does it give me The Rules of Differentiation to find the Derivative of a function.
As far as I am aware a Differential is NOT the same as a Derivative, but that a Differential is a part Derivative. That is to say every Derivative has a Differential as a part of it, but not every Differential is part of a Derivative.
TLDR; Why are they called Rules of Differentiation and not Rules of Derivation?
 A: Too long for a comment.
I am going to give you another example of weird usage of terms but that we use it in this way because those were the term that stuck.
In probability, there exists the concept of renewal theory (events that once they happen, the entire process starts from scratch). A typical example is a return to the origin by the random walk. Once the random walk reaches the origin, you can forget the entire past and assume the random walk started from scratch. Now, in renewal theory there are two types of events that are of interest, those that can happen infinitely many times and those that will happen (almost) surely a finitely number of times. The latter type of event are usually called "transitory" which makes perfect intuitive sense. Unfortunately, the former are called "recurrent" which is not quite right since the expected time between two "recurrent" events can be infinite! (In other words, the amount of time you would expect to pass between two successive occurrences of this "recurrent" event is likely to be very large which makes the term "recurrent", literally "occuring often or frequently", a really bad choice of terminology.) The better term "persistent" event never become popular even though the events really are "persistent" and not "recurrent" (they can happen again and again even though they don't happen often). And since I am touching this topic, the "recurrent" events are further classified into two: those with finite waiting time and those with infinite waiting time; those with finite waiting time are really "recurrent" in the English sense.
