According to Wikipedia:
The partial-derivative symbol is ∂. One of the first known uses of the symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation is by Adrien-Marie Legendre (1786), though he later abandoned it; Carl Gustav Jacob Jacobi re-introduced the symbol in 1841.
Here, we also find that:
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, dz & ∂z désigneront ou deux differences partielles de z,, dont une par rapport a x, l'autre par rapport a y, ou bien dz sera une différentielle totale, & ∂z une difference partielle. [Throughout this paper, both dz & ∂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 dz and ∂z will be employed as symbols of total differential, and of partial difference, respectively.]