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I was wondering today about why the word differentiable is used for describing functions that have a derivative or are differentiable.

Perhaps because originally one considered finite differences? But that seems somewhat not right, because roughly speaking a derivative measures not the difference $f(x+h)-f(x)$, but rather the ratio $(f(x+h)-f(x))/h$.

So, could people here shed light on why we use "differentiable"? Any pointers to academic / historical / etymological explanations are also welcome. Thanks!

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From the Earliest Known Uses of Some of the Words of Mathematics webpage:

DIFFERENTIAL CALCULUS. The term calculus differentialis was introduced by Leibniz in 1684 in Acta Eruditorum 3. Before introducing this term, he used the expression methodus tangentium directa (Struik, page 271). The OED has a nice quotation from Joseph Raphson’s Mathematical Dictionary of 1702: “A different way....passes....in France under the Name of Leibnitz's [sic] Differential Calculus, or Calculus of Differences.”

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    $\begingroup$ +1: Thanks for the link. Do you think that Leibniz used "differentialis" because essentially he might have been thinking in terms of "differences"? $\endgroup$ – user1709 Dec 29 '10 at 21:13

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