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."
