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Cajori, the website on Earliest Uses of Symbols of Calculus and many other sources claim that Lagrange introduced the notation $f'(x)$ for the derivative of $f(x)$ with respect to $x$. But I see Euler using it 1748 in Sur la vibration des cordes p. 78-79

[$\ldots$] pourvu que nous marquions le differentiel de la fonction $f{:}z$ par $dz \, f'{:}z$, & le differentiel de la fonction $f'{:}z$ par $dz \, f''{:}z$.

(He's using the alternative notation for "function application" $f{:}x$, instead of the nowadays common $f(x)$.)

Around this time Lagrange was 12 and, according to Wikipedia, showed no interest in mathematics prior to being 17. So it seems unlikely that Euler got it from Lagrange. On the other hand, Lagrange studied the works of Euler and it is reasonable to suspect that Lagrange took it from him.

Can someone confirm this state of affairs?

Michael Bächtold
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2 Answers2

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I agree, as there is further evidence that Lagrange got his primes from Euler:

1. Everyone since Cajori (1923, p. 6; 1929, p. 207) quotes Théorie des fonctions analytiques (1797) for the (sic) “new” notation $\ f'x,$ $\ f''x,$ $\ f'''x,$ $\ f^{\scriptsize{\mathrm{IV}}}x$. But Euler’s much earlier Institutionum calculi integralis vol. 3 (1770) had already systematically deployed his own $\ f'\!\!:\!x,$ $\ f''\!\!:\!x,$ $\ f'''\!\!:\!x,$ $\ f^{\scriptsize{\mathrm{IV}}}\!\!:\!x$ (see e.g. pp. 37-41, 45-65, 75-88, 93-103, 108-149, 234-237, 242-250, 262-268, etc). Notationally, Lagrange just dropped the evaluation colon.

2. Cajori (ibid.) adds that Lagrange wrote $\psi'x$ already in (1770, p. 275). But he had $\varphi't,$ $\varphi''t,$ $\varphi'''t,$ $\varphi^{\scriptsize{\mathrm{IV}}}t$ still earlier in (1766, pp. 206, 212, 275) and (1762, pp. 43, 63), and these two papers are both immediately preceded in the same journal (edited by Lagrange) by contributions of Euler, featuring $\Gamma'\!\!:\!u,$ $\ \Gamma''\!\!:\!u,$ etc. in (1766, pp. 13, 32, 62, 69, etc.) and $\varphi'z,$ $\ \varphi''z$, etc. in (1762, p. 9).

3. Cajori also quotes Jourdain (1913, p. 540) to the effect that Lagrange likely wrote the section of de Foncenex (1762, p. 321) where $\xi'x,$ $\xi''x,$ $\xi'''x,$ $\xi^{\scriptsize{\mathrm{IV}}}x\ $ “denote, as is the usual custom, the successive differences of $\xi x$ divided by $dx$”. But this follows the above-mentioned papers of Euler and Lagrange (same volume).

4. Anyway, as you found, Euler had used his own primes before, in (1749, p. 521) $=$ (1750, p. 79). He did again in (1755, pp. 212, 215), (1764, p. 256), (1766, pp. 195, 220, 250) — written in 1759 and featuring, p. 258, the remarkable sentence

les caracteres $\Phi$ & $\Psi$ marquent des fonctions quelconques régulieres ou irrégulieres, d’où par la différentiation on aura les fonctions dérivées $\Phi'$ & $\Psi'$,

(1767, pp. 316, 328, 332, 344), (1773, pp. 215, 224, 388, 395-399, 429, 436) which mostly drops the colon, etc. Moreover, published correspondence shows Lagrange aware of Euler’s papers (1750, 1762) and book (1770) “qui roule entièrement sur le calcul des fonctions”, by respectively October 1759, January or June 1760, and April 1771.

5. Nor were Euler and Lagrange alone in using prime notation: e.g. Monge (1773, p. 85), (1776, pp. 268, 288, 298), (1786, p. 22), (1787, pp. 87, 89, . . . , 172-185), (1801, Nº 8, 10, 14, 16, 33), Cousin (1777, p. 283), Marie (1793, pp. 8, 18), Bossut (1798, p. 427), . . .

So the “earliest uses” site, Cajori, wikipedia, et al., seem far from telling the whole story.

Euler titles (see Opera Omnia (2) vol. 10, 11 (1947, 1957) and (3) vol. 1 (1926)):
(1749) E119: De vibratione chordarum exercitatio.
(1750) E140: Sur la vibration des cordes.
(1755) E213: Remarques sur les mémoires précedens de M. Bernoulli.
(1762) E268: Lettre de M. Euler à M. de la Grange.
(1764) E287: De motu vibratorio cordarum inaequaliter crassarum.
(1766) E305: De la propagation du son.
(1766) E306: Supplément aux recherches sur la propagation du son.
(1766) E307: Continuation des recherches sur la propagation du son.
(1766) E317: Éclaircissemens sur le mouvement des cordes vibrantes.
(1766) E318: Recherches sur le mouvement des cordes inégalement grosses.
(1766) E319: Recherches sur l’intégration de l’équation $\smash{\bigl(\frac{ddz}{dt^2}\bigr)=aa\bigl(\frac{ddz}{dx^2}\bigr)+\frac bx\bigl(\frac{dz}{dx}\bigr)+\frac c{xx}z}$.
(1767) E339: Sur le mouvement d’une corde qui au commencement n’a été ébranlée que dans une partie.
(1767) E340: Éclaircissemens plus détaillés sur la génération et la propagation du son, et sur la formation de l’echo.
(1770) E385: Institutionum calculi integralis volumen tertium.
(1773) E433: Digressio de traiectoriis tam orthogonalibus quam obliquangulis.
(1773) E439: De chordis vibrantibus disquisitio ulterior.
(1773) E441: De motu vibratorio chordarum ex partibus quotcunque diversae crassitiei compositarum.
(1773) E442: De motu vibratorio chordarum crassitie utcunque variabili praeditarum.

Lagrange titles (see Taton (1974)):
(1762) T6: Nouvelles recherches sur la nature et la propagation du son.
(1766) T11: Solution de différens problèmes de calcul intégral.
(1770) T18: Nouvelle méthode pour résoudre les équations littérales par le moyen des séries.
(1774) T33: Sur une nouvelle espece de calcul, rélatif à la différentiation et à l'intégration des quantités variables.
(1797) T102: Théorie des fonctions analytiques.
(1801) T114: Leçons sur le calcul des fonctions.

Francois Ziegler
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    That find of Euler using $f:x$ is very nice. In particular this phrase suggests, that he had a clearer understanding of $f$ than some of his successors. – Michael Bächtold May 29 '18 at 14:01
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    Nice find Francois. – Mikhail Katz May 29 '18 at 18:49
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    It’s a mystery why this book of Euler is ignored so widely (e.g. by Cajori: totally) even though it went through 5 editions + German translation from 1770 to Opera Omnia (1914). (Also hard to fathom why Jourdain et al. keep belaboring de Foncenex’s primes, with Euler and Lagange’s right there in the same volume. Probably just shows how blessed — or cursed? — we are with easier access to most everything.) – Francois Ziegler May 30 '18 at 18:17
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    Interesting update. You're suggesting Lagrange might have gotten the "derived function" terminology from Euler? (That sentence of Euler is also remarkable since in the first part he calls $\Psi, \Phi$ characters, while in the second he calls $\Psi', \Phi'$ functions, subtly contributing to the popular $f=f(x)$ mix-up.) – Michael Bächtold May 31 '18 at 10:54
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    @FrancoisZiegler these findings raise another question for me: did Euler ever write $y'$ for $dy/dx$ when $y$ was a function of $x$? Or was this unfortunate (imo) invention due to Lagrange? – Michael Bächtold Jun 02 '18 at 07:45
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    Yes, I am. And yes, he did: E433, p. 215 (now added) has $d\mathrm X = \mathrm X'dx$, and with $\mathrm X=x^m$, $\mathrm X'=m.x^{m-1}$. – Francois Ziegler Jun 04 '18 at 19:12
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That's an interesting find but notice that Euler is still talking about differentials here. He seems to use $f'$ as somewhat ad hoc notation for an auxiliary entity used in describing the differential, rather than having the " $f\mapsto f'$ " picture in mind. Lagrange introduced the new concept of a "derived function" even if he borrowed this piece of notation from Euler.

Mikhail Katz
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    If you look in Eulers text a few lines above the quote, you find that he is using $f'$ and $f''$ in the same way we do it today. He even applies the chain rule correctly. In which sense do you find his $f'$ to be a different entity than our "derived functions"? – Michael Bächtold Jun 20 '17 at 14:27
  • Concerning Lagrange: I'm not sure he had a different picture in mind than Euler. The modern idea $f\mapsto f'$ considers $f$ as an object in itself, but it seems that prior to Frege, Dedekind and Cantor no one explicitly thought of $f$ that way. A function was defined to be a variable that changes whit some other variable, and not as the rule that expresses the dependence. See my other question – Michael Bächtold Jun 20 '17 at 14:29
  • Michael, Leibniz also applied the chain rule correctly. Yet neither Leibniz nor Euler had a notion of a "fonction derivee" (sorry about lack of diacritical symbols). I agree that Dirichlet, Dedekind, and others developed a more abstract notion of a function than Lagrange had. Yet historians generally do attribute the notion of "fonction derivee" to Lagrange, even though it may have had more limited scope than the notion developed in the 19th century. – Mikhail Katz Jun 20 '17 at 14:37
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    Euler certainly had the notion of derived function, even though he might not have called it that way. In his institutiones calculi differentialis 1748 he writes "For this reason if y is an algebraic function of x, then dy/dx is also an algebraic function of x." He goes on to compute dy/dx for all sorts of functions of x, not only algebraic ones. – Michael Bächtold Jun 20 '17 at 14:59
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    Michael, I share your enthusiasm for Euler's genius, as well as Leibniz's genius. Note that Leibniz also spoke about the differential ratio $\frac{dy}{dx}$. But your claim that Euler had the notion of derived function is unsupported. – Mikhail Katz Jun 20 '17 at 15:03
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    He explicitly calls dy/dx a function of x. Why is that no proof of the claim that he had a notion of derived function? – Michael Bächtold Jun 20 '17 at 15:05
  • As a side remark: I don't think Lagrange is less genius. – Michael Bächtold Jun 20 '17 at 15:08
  • I agree! Sorry, should have included him to begin with. Then again the list of all the geniuses in mathematics is likely to outrun the space limits of an HSM comment :-) – Mikhail Katz Jun 20 '17 at 15:12
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    Allow me to clarify what I find remarkable about Eulers use of the chain rule here: previously it had only been written as $dz/dx=dz/dy\cdot dy/dx$ while now he uses the $f$ and $f'$-notation, in a way that agrees with modern use! This may seem like a triviality, but contrast it with his first use of the $f$-notation in E045 (1734), where the same letter $f$ was used for every function he encountered, even if they were different. That use is not compatible with modern use of $f$. – Michael Bächtold Jun 21 '17 at 10:31