In fact, it’s his exact notation we generally use today for the derivative \frac\pi. Leibniz in particular blurred the distinction between infinitesimals and real numbers, treating them as one in the same. The question Newton and Leibniz were asking throughout their development of “infinitesimal calculus” was how a tiny nudge of dx affects the value of various functions y=f(x), in other words, a tiny nudge dy! An infinitesimal describes a tiny nudge to the value of x far smaller than the degree of precision being used in conventional calculations… infinitesimally small, so to speak. Rather than simply writing it for the sake of convention within a limit-based derivative or integral, though, this symbol came to represent a quantity itself. Rather than relying on the rigor of axioms and limit postulates to develop a fully consistent system from the ground up, Newton and Leibniz- the pioneers of modern Calculus’s main ideas- built their new language around a murky idea that came to be known as “the infinitesimal.” That may sound mysterious, but I’ll bet that the symbol below is familiar to even the most limit-rooted Calculus students. What do I mean by this? Like all concepts we treat as mathematical law today, Calculus was as fluid with its rules as the continuous change it described. Plus, I could rest assured that my methods were backed by the fathers of Calculus themselves. Sounds stupid, I know, but at the time I was engrossed in an alternate system that explained the central ideas of derivatives and integrals seamlessly without the minefield of epsilons and deltas that scare away half of the first-time students opening a Calc textbook for the first time. When I was learning Calculus on my own, I decided to skip limits.
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