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."