Calculus, the mathematical study of change, can be separated into two departments: differential calculus and integral calculus. Both deal with infinite sequences and series to define a limit. To produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus forming ideas and theories, but it was not until the late 17th century that the theories and concepts were specified. Originally called infinitesimal calculus, it meant creating a solution for calculating objects smaller than any previously known feasible measure through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but controversy appeared when both wanted sole credit for the invention of calculus. This article will show the typical reason why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system. Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse and Antiphon the Sophist, were the first to discover the basic elements that resulted in what we understand today and which formed the branch of mathematics called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of ​​a parabolic segment, as an example of the sum of an infinite series. He also used the method of exhaustion, invented by Antiphon, to approximate the area of ​​a circle, as an example of early integration. Realizing the Indian mathematicians, Aryabhata and...... middle of paper ......ocity. On the other hand, Leibniz had taken a geometric approach, basing his findings on the work of earlier thinkers such as Fermat and Pascal. Although Newton was the first to derive calculus as a mathematical approach, Leibniz was the first to widely spread the concept throughout Europe. This was perhaps the most conclusive evidence that Newton and Leibniz were both independent developers of calculus. Newton's timeline shows more evidence of the invention of calculus due to his refusal to use theories or concepts to prove his answers, while Leibniz promoted the ideas of other mathematicians to collaborate and bring together theorems for the application of calculus . The history of calculus developed as a result of sequential events, including many inventions and innovations, which led to thinking about the future in the development of the mathematical system.