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I am wondering why the notation $\frac{df}{dx}$ isn't used for partial derivatives, because it seems to me like someone could tell that it was a partial derivative if they knew that $f$ was a function of several variables, so no need to use a different notation.

HDE 226868
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Tdonut
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    This is not backed up by any historical research, but mathematically: $df/dx$ acts like a fraction (say, in the chain rule) but $\partial f/\partial x$ doesn't, so it's nice to have different notations. – Mark S. Nov 09 '15 at 22:11

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As this states,

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).

Throughout this paper, both $dz$ & $\partial z$ 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 $dz$ and $\partial z$ will be employed as symbols of total differential, and of partial difference, respectively.

Caritat wanted to differentiate (if you'll pardon the pun) between the total differential and the partial derivative, which have two completely different meanings (This is elaborated upon in an answer by Lost). Legendre and Jacobi later used the same $\partial$ for partial derivatives, and the notation stuck.

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