In a nutshell, the calculus was "discovered" or "invented" during 17th century independently by Leibniz and Newton who merged brilliantly various techniques developed since ancient Greece to solve geometrical problems.
Following the development of algebra during the Reanaissance and the pubblication of Descartes' Geometry in 1637, those methods were improved and new ones were discovered:
The "official" birth of the calculus must be dated with Newton (De analysi of 1669) and Leibniz (various Ms. of 1675) independent developments:
Leibniz's differential and integral calculus and Newton's fluxional calculus,
though different in many aspects, each involve a clear recognition of what we now call the inverse relationship between differentiation and integration. Moreover, both men worked out a system of notations, symbols and rules through which their methods could be applied in the form of algorithms performed on formulae, rather than in the form of geometrical arguments presented in prose with reference to figures.
See into: Ivor Grattan-Guinness, cit., Ch.2 Newton, Leibniz and the Leibnizian Tradition, by H.J.M. Bos, page 49-on