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Derivatives.
In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative. In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum distributed in 1636, Fermat introduced the concept of adequality, which represented equality up to an infinitesimal error term. This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation.
Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents.
Fundamental theorem of calculus.
The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time and Fermat's adequality. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.
James Gregory, influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus, that integrals can be computed using any of a function’s antiderivatives.
The first full proof of the fundamental theorem of calculus was given by Isaac Barrow.
Other developments.
One prerequisite to the establishment of a calculus of functions of a real variable involved finding an antiderivative for the rational function .
This problem can be phrased as quadrature of the rectangular hyperbola xy = 1. In 1647 Gregoire de Saint-Vincent noted that the required function F satisfied , so that a geometric sequence became, under F, an arithmetic sequence. A. A. de Sarasa associated this feature with contemporary algorithms called logarithms that economized arithmetic by rendering multiplications into additions. So F was first known as the hyperbolic logarithm. After Euler exploited e = 2.71828..., and F was identified as the inverse function of the exponential function, it became the natural logarithm, satisfying .

Shaded area of one unit square measure when x = 2.71828... The discovery of Euler’s number e, and its exploitation with functions ex and natural logarithm, completed integration theory for calculus of rational functions.
The first proof of Rolle's theorem was given by Michel Rolle in 1691 using methods developed by the Dutch mathematician Johann van Waveren Hudde. The mean value theorem in its modern form was stated by Bernard Bolzano and Augustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Important contributions were also made by Barrow, Huygens, and many others.