Partial derivative. The "curly d" was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in Memoire sur les Equations aux différence partielles, which was published in Histoire de L'Academie Royale des Sciences, pp. 151-178, Annee M. DCCLXXIII (1773). On page 152, Condorcet says:
Dans toute la suite de ce Memoire, $\text{d} z$ & $\partial z$ désigneront ou deux differences partielles de $z$, dont une par rapport a $x$, l'autre par rapport a $y$, ou bien $\text{d} z$ sera une différentielle totale, & $\partial z$ une difference partielle.
[Throughout this paper, both $\text{d} z$ & $\partial z$ will either denote two partial differences of $z$, where one of them is with respect to $x$, and the other, with respect to $y$, or $\text{d} z$ & $\partial z$ will be employed as symbols of total differential, and of partial difference, respectively.]
However, the "curly d" was first used in the form $\frac{\partial u}{\partial x}$ by Adrien Marie Legendre in 1786 in his Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations, Histoire de l'Academie Royale des Sciences, Annee M. DCCLXXXVI (1786), pp. 7-37, Paris, M. DCCXXXVIII (1788). On page 8, it reads:
Pour éviter toute ambiguité, je répresentarie par $\frac{\partial u}{\partial x}$, le coefficient de $x$ dans la différence de $u$, & par $\frac{\text{d} u}{\text{d} x}$ la différence complète de $u$ divisée par $\text{d} x$.
Legendre abandoned the symbol and it was re-introduced by Carl Gustav Jacob Jacobi in 1841. Jacobi used it extensively in his remarkable paper De determinantibus Functionalibus, Crelle’s Journal, Band 22, pp. 319-352, 1841 (pp. 393-438 of vol. 1 of the Collected Works).
Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristica $\text{d}$ differentialia vulgaria, differentialia autem partialia characteristica $\partial$ denotare.
The "curly d" symbol is sometimes called the "rounded d" or "curved d" or Jacobi’s delta. It corresponds to the cursive "dey" (equivalent to our d) in the Cyrillic alphabet.
Quindi, piuttosto che un $d$ corsivo, è una lettera corsiva (questa) dell'alfabeto cirillico.
