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When the action is: Taking the derivative

what verb should be used?

  • to differentiate

  • to derive

I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I right or has the word derive got the same meaning as differentiate? Or perhaps differentiate is not a proper English word...? If so, can anyone name a book or article where the writer(s) (preferably native English speaker(s)) use the word derive to mean differentiate? Or should we always stick to saying: "Take the derivative of..."?

Edit: So from what I can tell, the phrase: "Derive a method for differentiating this function and write down the resulting derivative.", can only have one meaning. XD

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    $\begingroup$ The correct verb is to differentiate. The corresponding noun is differentiation. The mathematical meaning of 'to differentiate' ca be found through google (it's no. 3) $\endgroup$
    – Danu
    Jul 10, 2014 at 11:48
  • $\begingroup$ I'm not 100% sure this is canonical, but you either take a derivative or differentiate. 'Derive' often means 'solve' or 'find a solution'. $\endgroup$
    – Alex
    Jul 10, 2014 at 11:49
  • $\begingroup$ Your summary sentence ("Derive how you should...") is a little awkward; more often that would be "Explain how you should..", or something like that. I recommend "Derive a method for differentiating this function and write down the resulting derivative." $\endgroup$ Jul 10, 2014 at 17:55
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    $\begingroup$ "Derive" and "differentiate" have different derivations, and it's important to differentiate between them ;-) $\endgroup$
    – David
    Jul 15, 2014 at 5:06
  • $\begingroup$ It probably comes about because “differential” can also be part of a car (making “differential equation” a misleading phrase for non-mathematicians). “derivative” is arguably a clearer word. The logic then goes that a derivative must have been derived from something. But surprisingly, it's not actually that common. I think, to a native speaker, “-tive” doesn't quite work to back-form a synonym for “differentiate”. $\endgroup$
    – mudri
    Jul 15, 2014 at 23:05

3 Answers 3

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In English, I've almost always heard mathematicians say "We now differentiate $f$ to get ...". Occasionally I've heard "derive," but in English (my native language!), that's generally used to mean "work out", as in "Ralph couldn't derive a proof of the intermediate value theorem from the information he had at hand." It's also used in generating one thing from others, as in "We can now derive the half-angle formulas from the addition formula by a clever substitution."

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    $\begingroup$ I had a pretty strong feeling that this would be the case, but I've seen people doing it wrong so often, that I began doubting. $\endgroup$
    – gebruiker
    Jul 10, 2014 at 11:53
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    $\begingroup$ I agree: I hear "derive" used for "differentiate" only by non-native speakers (of English, or of calculus). $\endgroup$ Jul 11, 2014 at 16:27
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That is a question for a native speaker, I fear.

In German both are used

  • to differentiate = differenzieren (determing the derivative)
  • to derive = ableiten -> Ableitung (derivative)

In English literature, I think I only saw differentiate for the operation.

In German you can use "Herleitung" to stress more that it is about taking conclusions. In English it is maybe "derivation".

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    $\begingroup$ The same in French : both words are used with the same meaning ($differencier$ and $dériver$). $\endgroup$ Jul 10, 2014 at 11:55
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    $\begingroup$ Same in Swedish $\endgroup$ Jul 10, 2014 at 14:06
  • $\begingroup$ but in german, do you obtain a derivative or a differential? $\endgroup$ Mar 20, 2021 at 18:54
  • $\begingroup$ @ErikKaplun I fear I do not get your question. Anyways: (de: Ableitung) = (en: derivative) e.g. $F'$, (de: Differential) = (en: differential) e.g. $dF$. $\endgroup$
    – mvw
    Mar 21, 2021 at 0:23
  • $\begingroup$ In French we have clearly two separate and interchangeable families: Process dérivation and différentiation (with a 't' not a 'c', in the latter case it means making a difference); Result dérivée and différentielle; Property dérivable and différentiable. Historical, concurrent work by different mathematicians. $\endgroup$
    – mins
    Apr 12, 2021 at 13:09
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In mathematical/Calculus sense only: to differentiate is the verb "to find or calculate the derivative" The noun is "the derivative" Non-calculus students assume that it's ok to say or ask "how do I derive this function?" NO! Calculus students ask, "How do I differentiate this function?" Or "How may I find the derivative of this function?"

Conclusion: right now, in the 21st century Calculus instructors use the verb = differentiate and the noun = derivative

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