 Registration time
 2009316
 Last login
 20111017
 Online time
 341 Hour
 Reading permission
 90
 Credits
 20682
 Post
 4440
 Digest
 0
 UID
 211674

The Greek mathematician Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together thereby finding the tangent to the curve.[7]
The Indian mathematicianastronomer Aryabhata in 499 used a notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[8] Manjula, in the 10th century, elaborated on this differential equation in a commentary. This equation eventually led Bhāskara II in the 12th century to develop the concept of a derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[8][9][10]
In the late 12th century, the Persian mathematician, Sharaf alDīn alTūsī, introduced the idea of a function. In his analysis of the equation x3 + d = bx2 for example, he begins by changing the equation's form to x2(b − x) = d. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value d. To determine this, he finds a maximum value for the function. Sharaf alDin then states that if this value is less than d, there are no positive solutions; if it is equal to d, then there is one solution; and if it is greater than d, then there are two solutions. However, his work was never followed up on in either Europe or the Islamic world.[11]
Sharaf alDīn was also the first to discover the derivative of cubic polynomials.[12] His Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions. For example, in order to solve the equation x3 + a = bx, alTusi finds the maximum point of the curve y = bx  x^3\,\!. He uses the derivative of the function to find that the maximum point occurs at \textstyle x = \sqrt{\frac{b}{3}}\,\!, and then finds the maximum value for y at \textstyle 2(\frac{b}{3})^\frac{3}{2}\,\! by substituting \textstyle x = \sqrt{\frac{b}{3}}\,\! back into y = bx  x^3\,\!. He finds that the equation bx  x^3 = a\,\! has a solution if \textstyle a \le 2(\frac{b}{3})^\frac{3}{2}\,\!, and alTusi thus deduces that the equation has a positive root if \textstyle D = \frac{b^3}{27}  \frac{a^2}{4} \ge 0\,\!, where D\,\! is the discriminant of the equation.[13]
In the 15th century, an early version of the mean value theorem was first described by Parameshvara (1370–1460) from the Kerala school of astronomy and mathematics in his commentaries on Govindasvāmi and Bhaskara II.[14]
In the 17th century, European mathematicians Isaac Barrow, 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, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.[15] Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."[16]
The first proof of Rolle's theorem was given by Michel Rolle in 1691 after the founding of modern calculus. The mean value theorem in its modern form was stated by Augustin Louis Cauchy (17891857) also after the founding of modern calculus. 
