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http://en.wikipedia.org/wiki/Calculus#History

in India, the mathematician Aryabhata in 499 made use of the infinitesimal and discovered the differential equation. Manjula in the 10th century, elaborated on Aryabhata's differential equations in a commentary. Bhaskara in the 12th century, developed a number of ideas that are foundational to the development of modern calculus, including the statement of the theorem now known as "Rolle's theorem", which is a special case of one of the most important theorems in analysis, the mean value theorem. He was also the first to develop the derivative and differential coefficient, and hence the first to conceive of differential calculus. Using these concepts, he found the differentials of the sine function and velocity of Earth's elliptical orbit around the Sun.

The 14th century Indian mathematician-astronomer Madhava, along with other mathematician-astronomers of the Kerala School, studied mathematical analysis, infinite series, power series, Taylor series, trigonometric series, convergence, differentiation, integration, term by term integration, numerical integration by means of infinite series, iterative methods for solutions of non-linear equations, tests of convergence, the concept that the area under a curve is its integral, and the mean value theorem, which was later essential in proving the fundamental theorem of calculus and remains the most important result in differential calculus. Jyeshtadeva of the Kerala School wrote the first differential calculus text, the Yuktibhasa, which also includes discoveries of integral calculus, and explores methods and ideas of calculus that were not discovered in Europe until the 17th, 18th and even 19th centuries. There is some evidence however, that these developments of calculus were transmitted to Europe via traders and Jesuit missionaries.