Supplementary mathematics/Derivative

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Derivative is the main idea of differential calculus, the first part of mathematical analysis that shows the rate of change of the function. The derivative, like the integral, is derived from a problem in geometry, i.e. finding the tangent line at a point of the curve.

The concept of the derivative was not established until the beginning of the 17th century, that is, before the French mathematician Pierre de Fermat determined the extrema of some special functions. Fermat found that tangent lines must be horizontal at points where the curve has a maximum or minimum. Therefore, it seemed to him that the problem of determining the extremum points of the function is related to solving another problem, that is, finding horizontal tangents. It was the attempt to solve this more general problem that led Fermat to discover some preliminary ideas of the derivative concept.

At first glance, it seemed that there is no relationship between the problem of finding the area under a graph and the problem of determining the tangent line to the curve at a point, but the first person who understood these two concepts, which are apparently far from each other, actually have a relatively close relationship with They also say that Isaac Barrow was Isaac Newton's teacher.

But the concept of the derivative in its present form was first developed independently by Newton in 1666 and by Gottfried Leibniz a few years after him. These two scientists, in the continuation of their work, again independently presented the second part of mathematical analysis, i.e. integral calculus, which is based on the act of integration.

Newton investigated the derivative from the kinematic reasoning method and from a physical point of view and used it to obtain the instantaneous speed. But Leibniz, with a geometrical point of view, used the derivative to obtain the coefficient of the tangent angle in the curves. Each of these two scientists used separate symbols to represent the derivative.[1][2]

The development of differential and integral calculus in later times is attributed to Augustin Louis Cauchy, Bernhard Riemann and the Bernoulli brothers, Jacob and Johann. Guillaume de l'Hôpital (French: Guillaume de l'Hôpital), a French scientist, published the first textbook on mathematical analysis in 1696 called "Analysis of infinitesimals for the examination of curves", which was actually a summary of the lessons given by Johann Bernoulli to The title of the teacher was written for him. In this book, the rule for solving ambiguity in the limit is also given using the derivative, which is known as Hopital's rule, but it actually belonged to Johann Bernoulli.