Leibniz based his approach to calculus on infinitesimals - numbers that are bigger than zero but smaller than 1/2, 1/3, 1/4, ... and so on. Many people were uncomfortable with these, so they figured out how to do calculus without infinitesimals. That's how it's usually taught now.

But you can do calculus with infinitesimals in a perfectly rigorous way... and in some ways, it's easier! Here's a free online textbook that teaches calculus this way:

• H. Jerome Keisler, Elementary Calculus, https://people.math.wisc.edu/~hkeisler/calc.html

The picture here is from this book. There's a tiny little infinitesimal number ε, pronounced 'epsilon'. And 1/ε is infinitely big! These aren't 'real numbers' in the usual sense. Sometimes they're called hyperreal numbers:

• Wikipedia, Hyperreal number, https://en.wikipedia.org/wiki/Hyperreal_number

You can calculate the derivative, or rate of change, of a function 𝑓 by doing

(𝑓(𝑥+ε)−𝑓(𝑥))/ε

and then at the end throwing out terms involving ε. For example, suppose

𝑓(𝑥) = 𝑥²

Then to compute its derivative we do

((𝑥+ε)²−𝑥²)/ε

Working this out using algebra, we get

2𝑥+ε

Then, at the end, we throw out the term involving ε and get

2𝑥

This is the rate of change of the function 𝑥².

In 1961 the logician Abraham Robinson showed that hyperreal numbers are just as consistent as ordinary real numbers, and that the two systems are compatible in a certain precise sense. In 1976 Jerome Keisler, a student of the famous logician Tarski, published this elementary textbook that teaches calculus using hyperreal numbers. And now it's free with a Creative Commons copyright!