# Differentiate and Integrate

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

1. the etymology of differentiate
2. the etymology of integrate
3. why we speak of a “derivative”, but we “differentiate” instead of “derive”

Please note that by “etymology” I mean the mathematical history 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?

Differentiate comes from the Latin calculus differentialis [differential method], coined by Gottfried Leibniz:

1684   G. Leibniz Acta Eruditorum 3 469   Ex cognito hoc velut Algorithmo, ut ita dicam, calculi hujus, quem voco differentialem, 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 call differential, 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. at Fluxions,   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.)