Calculus has been developed over the course of two centuries, from about 1700 till about 1900. First the overall concept and useful results were discovered without a firm foundation, then people worked for a long time on refining the details. Nowadays it’s usually taught the other way round - from the basics like the definition of real numbers to the results like the derivative and the integral. But historically those things have been usefully employed with enormous success in math and physics much earlier than being defined in the modern way.