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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?

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    $\begingroup$ While I agree that the rules really derive a new function (the derived function or derivative of the original function) it seems to be common usage to use differentiation for the verb and derivative for the noun. In other words, this is so because we use it that way. $\endgroup$
    – William M.
    Commented Feb 3, 2022 at 20:37
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    $\begingroup$ I don't know the answer to your question, and I imagine that History of Science and Mathematics might be a better place to ask, but the word "derive", in English (and in mathematical English) has other meanings, whereas "differentiate" doesn't really have other commonly used meanings in math (though there are other meanings). Thus we differentiate a function, using rules of differentiation, in order to find the derivative. $\endgroup$
    – Xander Henderson ♦
    Commented Feb 3, 2022 at 20:38
  • $\begingroup$ I 'differentiate' a function to obtain its 'derivative'. So, 'derivative' is a noun. I 'integrate' a function to obtain its 'integral'. Again, verb and noun. Some people use 'derive' and 'derivation' to mean 'differentiate' and 'derivative'. But, they are more general terms that mean 'work the problem' and 'the result of work'. Yes, the term 'differential' usually means something different. $\endgroup$
    – J126
    Commented Feb 3, 2022 at 20:44
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    $\begingroup$ Language is not as stiff as we might like. Words come into and go out of "fashion." Some expressions stick, some do not. Like, like, for example. Or "It is me" instead of the grammatically consistent "it is I." In short, language is determined by the speakers, not the grammarians. BTW: in Spanish, it's derivada, y derivar and integral e integrar, so a similar issue arises. Sigh. $\endgroup$
    – Matematleta
    Commented Feb 3, 2022 at 20:57
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    $\begingroup$ @Matematleta In French, as in Spanish, we say "dérivée" for "derivative" and also "dériver " (and not "différencier") for "to differentiate", though "dériver", like in English "to derive", is a polysemic term. $\endgroup$
    – Jean Marie
    Commented Feb 3, 2022 at 21:38

1 Answer 1

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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.

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