# When and why did $\frac{dy}{dx}$ become $\frac{d}{dx}y$?

It's obvious for us, that $$\frac{dy}{dx}$$ can also be written as $$\frac{d}{dx}y$$, but skimming through Leibniz or Eulers writings I couldn't see them write the latter. I speculate that this change happened out of laziness, when the expression for $$y$$ became very long, but I wonder if it was also related with a shift of perspective from treating $$d$$ as the main operator (Leibniz and Euler) to treating $$d/dx$$ as main one (maybe starting with Lagrange?).

(If this question seems to be about a triviality, consider writing $$\frac{\log}{\log 2}x$$ or $$(\log/\log 2) x$$ instead of $$\frac{\log x}{\log 2}$$ in an article or in front of students.)

• @Namaste Sure, but what you write is not specific to $d/dx$. It applies in general to operators. Should we write $\cos \phi$ or $\cos (\phi)$? $dx$ or $d(x)$? Historically it was without parenthesis, unless the parenthesis were necessary for disambiguation, as in $\cos xy$ or $dxy$. Even $f(x)$ was originally written without parenthesis. – Michael Bächtold Oct 4 '19 at 11:23

If you mean conceptually, I remember reading that Arbogast “was the first writer to separate the symbols of operation from those of quantity.” Among those he influenced were Babbage, Herschel, Peacock, in whose edited translation of Lacroix that phrase is used to justify writing Taylor’s formula for $$u=f(x)$$ in the form $$\Delta u =\bigl(e^{h\frac d{dx}}-1\bigr)u \tag{*}$$
(1816, see pp. 486–489) — an improvement over the rather clumsy $$\Delta u =e^{\frac{du}{dx}h}-1$$ of Lagrange (1774, p. 195), Lacroix (1800, p. 13), Brinkley (1807), or Laplace (1820, p. 41). (Arbogast wrote it $$\Delta u=(e^{h\partial.}-1)\times u$$ (1800, p. 350), and Lacroix adopted (*) in his second edition (1819, p. 61).) The ability to meaningfully write such formulas may well be the “why” you are after.
Edit: As noted today on MO, Fourier (maybe first?) uses a “bare” $$\smash{\frac d{dx}}$$ in Théorie analytique de la chaleur (1822, Art. 399-405, 410-414, 419-422,...).
• Yes, I just meant typographically in the particular case $dy/dx$ vs. $d/dx\; y$. But your answer is good since it's more general and gives me pointers to search for the particular case. – Michael Bächtold Oct 15 '18 at 18:57
• @MichaelBächtold Oh right — Babbage et al. (pp. 486, 492) never seem to apply a “bare” $d\,/\,dx$. But their student De Morgan does (1836, pp. 52, 53, 83); earlier, Hamilton (1831, pp. 102, 104, 106; 1834, p. 261):$$\frac d{dt}\frac{δ\mathrm T}{δ\eta'}-\frac{δ\mathrm T}{δ\eta}=\frac{δ\mathrm U}{δ\eta}.$$ – Francois Ziegler Oct 16 '18 at 6:12