The History of Derivatives

The derivative in mathematics refers to the exact rate at which one quantity fluctuates with respect to another. In fact, in geometrical terms, the derivative is simply the slope or gradient of a curve at a particular point on the graph. It can also be referred to as the tangential slope to the curve at that particular point. The derivative is an integral element in differential calculus.

The inventor of Modern Calculus was Sir Isaac Newton, a mathematician cum physicist, who discovered it at around 1665. He came to a realization that the mathematics at the time was insufficient to solve particular problems and thus he came up with these new mathematical elements. Approximately around the same era, another mathematician known as Goltfried Leibnez came up with ideologies similar to Newtons. Specifically, Newton was intrigued to understand the calculation of the velocity of an object in any particular instance. He was keen and observant towards an apple falling and hitting someone on the head if they are sited under the tree, and he wanted to know the velocity of the apple just before impact (Goldstine, Herman Heine).

Over time, Leibniz and Newton worked hard to complete three significant developments in Calculus. Initially, they came up with an explanation of the algorithmic process in both differentiation and integration. Leibniz analyzed integration as an addition of infinite amounts in accumulated regions thus creates a reference towards the concept of area calculation as it relates to the integral definition. On the other hand, Isaac Newton, applied geometry to elaborate on the relationship between distance, acceleration, and velocity as the most important factor towards understanding the relationship between the integral and the derivative. Specifically, velocity is the derivative of distance while acceleration is velocitys derivative. Cauchy, another scientist, came up with the definition of a definite integral using limits while in the 19th century, Denis Poisson Simeon, defined the definite integral as the difference between anti-derivatives at their endpoints (Tenreiro, M. J. et al.). For instance, [F (d)-F(c)], where the endpoints are d and c. These two theories describe the first fundamental calculus theorem.

Afterward, these two scientists also recognized the applicability of both integration and differentiation which was initially discovered by Fermat. In fact, in as much as the predecessors of Newton and Leibniz considered solutions towards tangent and area problems, they were unsuccessful in their quest to apply Calculus as a general tool for solving mathematical problems. Thirdly, Leibniz and Newton were the pioneers of an explicit pronouncement and proof that integration and differentiation were in fact, inverse processes.

Another influence of Leibniz in the development of the derivative was his invention of the notation commonly used by students in Calculus until today. He came up with the symbol; o to refer to an infinite sum. He also came up with the integral or rather the summation of numerous infinitely minute areas denoting the area beneath the function y, as oy (dx).

It is, however, noteworthy that these two pioneers of calculus never worked in unison as each of them had their unique approach towards the subject. Eventually, they spent the latter part of their lives arguing about who introduced new methods to solve calculus problems, to the tune of accusing each other of plagiarism.

It is important to note that even though the names Leibniz and Newton were key figures into the introduction of the modern calculus, they were not responsible for the introduction of calculus as the core development of Calculus had already been introduced by Fermat among others (Loverro, Adam).

Works Cited

Goldstine, Herman Heine. A History of the Calculus of Variations from the 17th through the 19th Century. Vol. 5. Springer Science & Business Media, 2012. Print.

Loverro, Adam. "Fractional calculus: history, definitions, and applications for the engineer." Rapport technique, University of Notre Dame: Department of Aerospace and Mechanical Engineering (2004): 1-28. Print.

Tenreiro Machado, J., Virginia Kiryakova, and Francesco Mainardi. "A poster about the old history of fractional calculus." Fractional Calculus and Applied Analysis 13.4 (2010): 447-454. Print.