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

Question

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?

Answer 1

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:

http://iupap.org/wp-content/uploads/2014/05/A4.pdf

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 This standard has been superseded by ISO 80000-2, which maintains the use of the upright d for the differential operator.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.

Answer 2

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 $\text{d}$.

Answer 3

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