The practice of calculus dates all the way back to ancient Egypt, where in about 1800 BC, and Egyptian mathematician successfully calculated the volume of a pyramidal frustum. However, the actual discovery of calculus is attributed to two men, Isaac Newton and Gottfried Willhelm von Leibniz. The English and German mathematicians are both accredited with the discovery of calculus, despite the fact that Newton discovered it 8 years prior to Gottfried. At the time, calculus was known as infinitesimal calculus due to the fact that it dealt with infinitely small but still nonzero. They used these values due to how convenient they were when it came to their calculations and derivations, but in modern practice, we find that real numbers are much more logical, as was the basis of the criticism formed by Lord Bishop Berkley.
Calculus was originally used in the process of finding the area under a curve or to get the total maximum of quantities. It was also applicable to the basis of problems in physics and astronomy that defined its origins. There are two main exercises in calculus; differentiation, which was used for the calculations of velocity and acceleration, the slope of a curve, and optimization, and integration, which worked with calculations of area, volume, arc length, center of mass, work, and pressure. Once the general practice of calculus had begun, it proceeded into the developmental stage in which the Newton and Leibniz really brought all the foundations together into one general study. Their early practices did not always prove to be logical, and it was during the period after, known as the Rigorization, that mathematicians brought calculus to a logical and practical foundation.
It wasn't until the mathematicians during the Rigorization period had come about that mathematicians like Augustin Louis Cauchy, Karl Weierstrass, and Georg Riemann realized that instead of focusing on infinitely small numbers was not a logical practice, and replaced it instead with the idea of limits, or the notion of numbers coming close to others.