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When did the word "derivative" come into use in calculus, and why?

As in Can the word "derive" be used to mean "take the derivative of"? the word "derivative" in normal English means "stemming from". But $\int f$ "derives" from $f$ just as much as does $f'$, and $f'$ "integrates" information from $f$ just as much as does $f'$. So who decided that a ratio of fluxions should be called the derivative, and why?

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You mean to ask what the derivative of "derivative" is? – fretty Jan 6 '14 at 18:10
@fretty Haha. Yes. From where did "derivative" derive? – isomorphismes Jan 7 '14 at 0:31

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The word originates from the French noun "Dérivée" (a feminine noun) introduced by Lagrange in the 18th century. One sometimes speaks of "deriving" a function but more commonly of "differentiating" it, because the term "derive" is commonly used in the logical sense of "obtaining as a consequence". To answer your question specifically, it seems to have been Joseph-Louis Lagrange who "decided" this.

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For my two cents' worth I must say that, although it may be clear what is meant when one speaks of "deriving" a function, this usage will make most mathematicians wince. The correct word is differentiate as you mention. Using "derive" in this way is akin to speaking of "timesing" two numbers instead of multiplying them. Please let's not encourage this slangy misuse! – MPW Jan 6 '14 at 18:07
Agreed. It is not so much "slangy" as translated (or adopted) from the French. – user72694 Jan 6 '14 at 18:11
Awesome, thanks! I was going to put Lagrange's name in the question because CTRL+F://derivative in brings his name into view—but it seemed too distant a connection to work as a guess! – isomorphismes Jan 7 '14 at 0:25
@MPW I agree that "differentiate" captures the sense of $f(x-h)-f(x)$ much better—it sounds almost like "difference", which is good. But then the question arises, why use "derivative" for the infinitesimal version instead of just "differences" (or "self-differences")? As notes, at some point we started dropping "infinitesimal" from "the infinitesimal calculus"—a disservice to all the other creative people who invented other interesting calculi! – isomorphismes Jan 7 '14 at 0:28
Leibniz used the term "differential" to emphasize the transition from finite differences to infinitesimal differences. His $dx$ and $dy$ and their properties were definitely inspired by finite differences and their properties, but they were not the same thing. That's why he introduced a new term. As far as calculus is concerned, what is dropped is not merely the word "infinitesimal" but infinitesimals themselves, as you may find out to your chagrin if you go to college. This was supposed to be a humorous remark :-) – user72694 Jan 7 '14 at 13:47

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