# $\frac{dy}{dx}$ versus $\frac{{\mathrm d}y}{{\mathrm d}x}$

When I first learned calculus a few decades ago, the books I read used italicized letter "d"s in derivatives (like this: $\frac{dy}{dx}$). But a few years ago, I started seeing upright "d"s (like this: $\frac{{\mathrm d}y}{{\mathrm d}x}$). When did this way of writing derivatives start?

• Perhaps 1992 (ISO-31); see also ISO 80000-2. Related TUGBoat article (1997). Basically $\mathrm d$ can't be a variable if it's upright, because variables are in italics. It's a question of consistency trumping beauty/tradition, somewhat like the fate of Pluto. Makes me sad. :( – Michael E2 Nov 20 '17 at 19:58
• I think the answer by @juan is sufficient? Also related: tex.stackexchange.com/questions/14821/… – Michael E2 Nov 21 '17 at 21:28
• In current mathematical publishing, you find predominantly italic d in the US and Roman d in Europe. – Gerald Edgar Nov 22 '17 at 2:00
• Similarly, in Europe you see $\mathrm{e}^x$ and not $e^x$ for the exponential function, and $\mathrm{i}$ and not $i$ for $\sqrt{-1}$ – Gerald Edgar Nov 22 '17 at 2:02
• My facetious answer is that "this started when people were able to squander time fooling with fonts, rather than content". For some period of time, I myself did go along with such distinctions, but ... I recovered. :) – paul garrett Mar 1 '18 at 2:02

The earliest use of upright d to indicate a differential or derivative operator as a standard practice I have found is from the International Union of Pure and Applied Physics (IUPAP), 1987:

This formed the basis for International Organization for Standardization's ISO 31 (1992) standard:

https://en.wikipedia.org/wiki/ISO_31

A discussion of this standard may be found in this 1997 TUGboat article. This standard has been superseded by ISO 80000-2, which maintains the use of the upright d for the differential operator.

Despite the claim that "the notations used in mathematics and science textbooks at schools and universities follow closely the guidelines in this standard," it is clear to me that the notation $dx$ is still widely followed in mathematics textbooks and journals in the US. Perhaps it is different in physics, which seems to be a source for the change in notation.

• FYI, the earliest appearance of this notation I know of is the 1966 book Calculus of Residues by Dragoslav S. Mitrinovic. However, this issue is not something that I've been keeping track of, so this reference may be a weak lower bound. I only remember this reference because a few months ago I made some notes from a library copy of this book, and my notes include the fact that the notation d$x$ is used in integrals, which I thought was interesting for a 1966 book. – Dave L Renfro Nov 22 '17 at 12:48
• @DaveLRenfro Yes, thanks. It seems to be a thing with that author: 1962, 1964, 1965. Surprising, if he did it on his own, but someone started it, I suppose. – Michael E2 Nov 22 '17 at 14:15
• The idea that d in d$x$ represents a differential operator is rather ahistorical. The notation predates such interpretations by about 200 years, for Leibniz it represented an infinitesimal length. Even Cartan, who invented the differential as an operator and $dx$ as a 1-form, still wrote it $dx$. – Conifold Nov 25 '17 at 4:15
• @Conifold do you have a reference of where Leibniz interprets d by itself as a length? (As opposed to interpreting dx as a length.) Also I assume that the reading of $dx$ as "the differential of $x$" is quite old, or even starts with Leibniz? – Michael Bächtold Mar 1 '18 at 8:12
• @MichaelBächtold "it" in the comment refers to $dx$, not $d$, "differential" was the infinitesimal increment, and Leibniz used $\omega$, $l$ and $x/d$ for it before settling on $dx$. – Conifold Mar 1 '18 at 8:27

Standards bodies recommend using roman font for mathematical operators and italic font for physical quantities or variables. The $\mathrm{d}$ in the derivative is an operator ("differential of"), therefore it has to be written in roman font as $\mathrm{d}x$, whereas a variable $d$ like in $y= ax^4 + bx^3 + cx^2 + dx$ is written in italic.

This convention also applies to other variables/operators. For instance a delta in roman font usually denotes the operator of small variation, whereas a delta in italic font usually means the decay coefficient. I don't know when those recommendations did start.

References:

On the use of italic and roman fonts for symbols in scientific text

Typefaces for Symbols in Scientific Manuscripts

• Note your physics.nist ref cites ISO-31 (1992), which is the earliest reference I know. I believe that was wehn the roman d became a standard, but I can't say that I know that. In the TeX Book (Knuth, 1984/86), uses the traditional $dx$. Knuth studied typesetting for the development of TeX, so I think one might conclude that the use of ${\mathrm d}x$ is not a standard before that. – Michael E2 Nov 21 '17 at 21:22
• @MichaelE2 Does Knuth TeXbook represent standard usages? I don't know if Knuth studied typesetting only in mathematical literature and ignored physicist/engineering literature (where standards are often different) and I don't know if Knuth studied European mathematical literature or only American literature when developed typesetting rules for Tex. Note as well that Knuth is inconsistent, because in the TeXbook he uses upright font for $\exp x$ (via the command \exp), but italic for its shorthand $e^x$. You find often $\mathrm{e}^x$ on European mathematical literature. – juanrga Nov 29 '17 at 20:36
• Upright for "exp" and italic for "e" is traditional. When you have several letters in a row forming a small block, the tradition has been to set them in the upright font. For $e^x$, it is viewed as basic algebra, like $2^3$, $b^x$, $x^n$, etc. One can distinguish $e$ from $b$ as a particular constant and decide it's better to set them in distinct types. I've seen $e$ both ways throughout the world, and some journals seem to accept both and leave it to the authors, including European ones. In math., it seems italic $e$ and $dx$ are more common as especially as you go back in time. – Michael E2 Dec 17 '17 at 15:25
• As for Knuth, AFAIR, he was using a TeX precursor in the late 70s, just about the time the New York Times stopped typesetting the newspaper. I don't know what printing communities he studied or how thoroughly, but certainly it was pre-TeX :), likely was mainly mathematical, probably mostly in US/Europe (many leading journals in math are European and well-known to US academics and always have been), and included hand-set type. I think ISO-31 & the LaTeX/AMSTeX control sequence for upright d are contributing to the spread of the upright d/e. Hard to know how typesetting will evolve. – Michael E2 Dec 17 '17 at 15:29

One already finds upright d's in Lacroix's Traité élémentaire de calcul différentiel et de calcul intégral (1802). I don't now if this is the earliest, but it is interesting to note that the first edition of his "non-elementary" Traité du calcul différentiel et du calcul intégral (1797) has italicized d's, while the second edition from 1810 has upright differentials.

Added: As noted in the comments by Francois Ziegler, Lacroix’s $$\mathrm d$$’s are predated by Lagrange (1762), L’Huilier (1795).

A somewhat tangential update: I was reading Euler's Institutiones Calculi Differentialis (1755; 1790) and found the following nice passage which could have influenced the change from $$d$$ to $$\mathrm{d}$$.

119. It should be kept in mind that the letter $$d$$ here does not denote a quantity, but is used as a symbol to express the word differential, in the same way that the letter $$l$$ is used for the word logarithm when the theory of logarithms is being discussed or the symbol $$\smash\surd$$ is used to denote the root. Hence $$dy$$ does not signify, as it usually does in analysis, the product of two quantities $$d$$ and $$y$$, but must be read as the differential of $$y$$. In a similar way, if we write $$d^2 y$$, this is not the square of a quantity $$d$$, but it is simply a short and apt way of writing the second differential. Due to this use of the letter $$d$$ in differential calculus, one should — in order to avoid confusion — not use it to denote a quantity in calculations when many different quantities occur, just like we avoid the letter $$l$$ to designate a quantity in calculations where logarithms occur. It would be desirable if the letters $$d$$ and $$l$$ were altered to a different appearance, lest they be confused with other letters of the alphabet that are used to designate quantities. This is what has happened to the letter $$r$$, which first was used to indicate a root; before it was distorted to $$\smash\surd$$.

And here's a quote from Cajori's A history of mathematical notation (1928) § 595.:

It looked indeed as if the different mathematical architects engaged in erecting a proud mathematical structure found themselves confronted with the curse of having their sign language confounded so that they could the less readily understand each other's speech. At this juncture certain writers on the calculus concluded that the inter­ests of their science could be best promoted by discarding the straight­ letter $$d$$ and introducing the rounded $$\partial$$. Accordingly they wrote $$\partial y$$ for the total differential, $$\frac{\partial y}{\partial x}$$ for the total derivative, and $$\left(\frac{\partial y}{\partial x}\right)$$ for the partial derivative. G. S. Klügel refers to this movement when he says in 1803$$^1$$: "It is necessary to distinguish between the symbol for differential and that for a [finite] quantity by a special form of the letter. In France$$^2$$ and in the more recent memoirs of the Petersburg Academy, writers$$^3$$ have begun to designate the differential by the curved $$\partial$$." So Klügel himself adopts this symbolism from now on for his dictionary. Euler's Institutiones calculi integralis, which in its first edition of 1768-70 used the straight $$d$$, appeared in the third edition of 1824 with the round $$\partial$$, both for the total differential and the total derivative. This same notation is found in J. A. Grunert's calculus$$^4$$ of 1837. However, the movement failed of general adoption; for some years both $$d$$ and $$\partial$$ were used by different writers for total differentiation.

$$^1$$: G. S. Klügel, Mathematisches. Wörterbuch, 1. Abth., 1. Theil (Leipzig, 1803), art. "Differentiale."

$$^2$$: In France, Le Marquis de Condorcet had used the $$\partial$$ in writing total deriva­tives in his Probabilité des décisions, in 1785.

$$^3$$ The Nova Acta Petropolitana, the first volume of which is for the year 1785, contain the rounded $$\partial$$.

$$^4$$ Johann August Grunert, Elemente der Differential und Integralrechnung (Leipzig, 1837).

But I couldn't find Cajori address the distinction $$d$$ vs $$\text{d}$$.

• Wow! I didn't know the upright d notation was that old. – Joel Reyes Noche Mar 1 '18 at 1:10
• I believe that systematic use of an upright $\mathrm d$ (along with $d$, $δ$, etc.) was started by Lagrange (1762, p. 230) — explained p. 174: “je dois avertir que (...) j’ai introduit dans mes calculs une nouvelle caracthéristique $δ$. Ainsi $δZ$ exprimera une différence de $Z$ qui ne sera pas la même que la $dZ$, mais qui sera cependant formée par les mêmes régles”. – Francois Ziegler Mar 1 '18 at 6:09
• Also: L’Huilier (1795) used $\mathrm dx$ throughout. Euler’s Integral calculus switched from $dx$ to $∂x$ between the first (1768) and posthumous second (1792) edition. Gothic books like Kästner (1761) had all math upright. Rondet (1735) saved $d$ by using $\mathrm d$ for coefficients: $\smash{az+bz^2+cz^3+\mathrm dz^4+ez^5}$... – Francois Ziegler Mar 1 '18 at 6:10
• @FrancoisZiegler, comments are temporary. I suggest that you write your two comments above as an answer so that they become more permanent. – Joel Reyes Noche Mar 1 '18 at 7:43
• @Michael Nice quote (also relevant to this question) in the update. – Francois Ziegler Oct 22 '18 at 16:41