Origin of the concept of derivative of a function

The concept of function that may seem simple today is the result of a slow and long historical evolution that began in antiquity when, for example, the Babylonian mathematicians used square and cubic root tables and tables or when the Pythagoreans tried to relate the height of the sound emitted by strings subjected to the same tension with their length. At this time the concept of function was not clearly defined: the relationships between variables emerged implicitly and were described verbally or by a graph.

Only in the century. XVII, when Descartes and Pierre Fermat introduced Cartesian coordinates, it became possible to transform geometric problems into algebraic problems and to study functions analytically. Mathematics thus receives a major boost, notably in its applicability to other sciences - scientists begin, from observations or experiments, to seek to determine the formula or function that relates the variables under study. From here the whole study develops around the properties of such functions. On the other hand, the introduction of coordinates, besides facilitating the study of known curves, allowed the "creation" of new curves, geometric images of functions defined by relationships between variables.

It was while devoting himself to the study of some of these functions that Fermat realized the limitations of the classical concept of tangent line to a curve as one that found the curve at a single point. It has thus become important to reformulate such a concept and find a process of plotting a tangent at a given point - this difficulty has been known in the history of mathematics as the "Tangent Problem."

Fermat solved this difficulty in a very simple way: to determine a tangent to a curve at a point P considered another point Q on the curve; considered the line PQ secant to the curve. He then slid Q along the curve toward P, thereby obtaining straight lines PQ that approached a line t to which Fermat drew the tangent line to the curve at point P.

Fermat noted that for certain functions, at points where the curve assumed extreme values, the tangent to the graph must be a horizontal line, since when comparing the value assumed by the function at one of these points P (x, f (x)) with the value Assuming at the other point Q (x + E, f (x + E)) near P, the difference between f (x + E) and f (x) was very small, almost zero, when compared to the value of E, difference Thus, the problem of determining extremes and determining tangents to curves becomes closely related.

These ideas were the embryo of the concept of Derivative and led Laplace to consider Fermat "the true inventor of differential calculus." However, Fermat was not properly rated and the concept of boundary was not yet clearly defined.

In the sixteenth century, Leibniz handcuffed Infinitésimal Calculus, introducing the concepts of variable, constant, and parameter, as well as the notation dx and dy to designate "the smallest possible differences in x and y." From this notation comes the name of the branch of mathematics. known today as "Differential Calculus".

Thus, although only in the nineteenth century did Cauchy formally introduce the concept of limit and the concept of derivative from the 19th century. With Leibniz and Newton XVII, Differential Calculus becomes an increasingly indispensable instrument for its applicability to the most diverse fields of Science.

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