Further to my last question about the history of calculus terms, I am wondering about

- the etymology of
differentiate- the etymology of
integrate- why we speak of a “derivative”, but we “differentiate” instead of “derive”
Please note that by “etymology” I mean the

mathematicalhistory of these terms.This may get answered

en route, but I am specifically interested in what (if any) relationship there may be between the ideas of “differentiating” (breaking apart) and “integrating” (putting together), and the fact that these two operations do in fact “undo” each other in mathematics, which is the essence of the Fundamental Theorem of Calculus.

EDIT:It just occurred to me that while “derivative” would seem to go with “derive”, “differentiate” would go with “difference”, and in fact “difference equations” were used by Leibniz when he developed calculus (although I don’t know if he called them as such) – can anyone verify this?

**Answer**

* Differentiate* comes from the Latin

*calculus differentialis*[differential method], coined by Gottfried Leibniz:

1684 G. Leibniz

Acta Eruditorum 3469 Ex cognito hoc velutAlgorithmo, ut ita dicam, calculi hujus, quem vocodifferentialem, omnes aliae aequationes differentiales inveniti poſſunt per calculem communem, maximaeque & minimae, itemque tangentes haberi[Just by knowing the

algorithm, as I call it, of this method, which I calldifferential, all other differential equations can be solved by a common method, and maxima and minima, and tangents too, can be found]

The name comes from the infinitesimal differences (*dx*, *dy*, etc., in Leibniz’ notation, also introduced in the 1684 paper) which are the basis of the method.

In English, the differential calculus was known at first as the *method* or *doctrine of fluxions* (Newton’s terminology). Thus the *OED*‘s first citation for *differential* in English is:

1702 J. Ralphson

Math. Dict.atFluxions, A different way..passes..in France under the Name of Leibnitz’s Differential Calculus, or Calculus of Differences.

(If you have more of these kinds of question, you’ll find Jeff Miller’s site "Earliest Known Uses of Some of the Words of Mathematics" a very helpful resource.)

**Attribution***Source : Link , Question Author : user76407 , Answer Author : Gareth Rees*