My sense is that Franklin was a scientist more like Faraday than Maxwell. Given also that Calculus was fairly new when Franklin was in school (and as far as I know, Benjamin Franklin did not get very far in school anyway) how likely was it that he could use Calculus or even really knew what it was?
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Apparently, no. Boman's biography Benjamin Franklin's Numbers specifically focuses on Franklin's mathematical activities, and calculus is not among them. He was deep into magic squares, and anticipated some basic ideas of statistics and population dynamics. In a 1772 letter to Joseph Priestley he described what he called Moral or Prudential Algebra for decision making, but that was more what is now called cost-benefit analysis. In Keith Devlin's description, the closest he came to calculus topics is predicting exponential growth of populations before Malthus in a two-page essay with no formulas:
"Franklin had a lifelong interest in magic squares... Now, to the outsider these look like just doodling with numbers but, in fact, to construct one of those things, you have to get deep into the mathematics and into the patterns. And Franklin did, in his lifetime, spent a lot of time constructing these things and you can only do that if you have a certain deep feeling for numbers.
The other example is what we find in his political and business writings. He is very early, much early than many other people in realizing that you could use what we would now call basic statistics in order to make political and business decisions.
... he wrote an article... in 1751 called Observations Concerning the Increase of Mankind and the Peopling of Countries. And that was really one of the first ever works in what we now call demographics, using mathematical techniques to look at how populations grow and how people move and how societies develop. In fact, Franklin was the first person who speculated that populations probably increase exponentially. Now, we always associated that with Thomas Malthus, who was the one that demonstrated that. But, in fact, Malthus had already read Franklin's work and cited it when he did his work."
Although Franklin did not do calculus he knew of it at least since 1725 at the age of 19. According to Bunker's Young Benjamin Franklin, when in London he met Pemberton, well versed in calculus and Newtonian physics, who was then preparing a new edition of Principia. Pemberton himself wrote an essay attempting to use Newtonian mechanics to explain the functioning of muscles, which Franklin read and which stimulated his interest in experimental science. A personal meeting with Newton, promised by Pemberton, never came to pass, but Franklin's interests shifted away from metaphysics, and four decades later he ordered a portrait of himself with Newton in the background.
In 1754 Franklin became one of the trustees of the Logan library established by the sons of late James Logan "a classicist who in the margins of his books crossed swords with greatest European editors, and a scientist who described the fertilization of corn by pollen, understood and used the new inventions of calculus, wrote on optics, and made astronomical observations, the Quaker virtuoso brought books to feed the wide-ranging appetite of his mind", see At the Instance of Benjamin Franklin. But they add:
"Although Franklin in his promotional tract for the establishment of a college in Philadelphia had described Logan's library as a valuable book resource available to professors and students, use was seldom made of the scholarly works in the collection. In the eighteenth century there was furthermore little interest in the classics and advanced mathematical sciences on the part of merchants and artisans in Philadelphia."
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$\begingroup$ Interesting. I would guess that a decent number of math faculty (although I am not sure in 18th century the idea of professor of math existed or corresponded to modern professor) might not have known much calculus in the mid 1700s, even in England or Germany. I note that Harvard only got a microscope for its school of medicine in 1860 or so, if that is related. Still: even if BF could not take a derivative I would bet he knew that Newton had developed calculus. $\endgroup$ Commented Dec 26, 2019 at 5:35
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1$\begingroup$ @releseabe He certainly knew that. When in London at 19 (1725) he met Pemberton, a master of calculus and an editor of Principia, and asked to meet Newton in person (unsuccessfully). Later he ordered a portrait of himself with Newton in the background. But his interest was in experimental science, not theory. Bunker's Young Benjamin Franklin has many interesting biographical details. $\endgroup$– ConifoldCommented Dec 26, 2019 at 6:02
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$\begingroup$ Sad that he did not meet Newton. But: Harrison met Newton and Harrison met Geo III who probably met Victoria (as an infant) and during the 20th century many people were around who met Victoria. Newton in 1725 had many health problems and in 1725 who the heck knew who BF was?? Kind of presumptuous of BF. But I think Newton would have not clearly understood the difference between theory and experiment -- pure math was a 19th century idea with some exceptions -- I guess number theory was always "pure." $\endgroup$ Commented Dec 26, 2019 at 6:19
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1$\begingroup$ @releseabe See edit. Bunker agrees with you on the meeting. Getting to know Lyons and Pemberton was more important in turning him from metaphysics to science. But Franklin took to that side of it that had to do with figuring out phenomena and staging experiments rather than writing formulas. Newton and Pemberton did both. $\endgroup$– ConifoldCommented Dec 26, 2019 at 6:39