![]() The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716), who provided independent and unified approaches to differentiation and derivatives. The Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), was the first to discover the derivative of cubic polynomials, an important result in differential calculus 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. The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114–1185) indeed, it has been argued that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem". The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 AD, when the astronomer and mathematician Aryabhata (476–550) used infinitesimals to study the motion of the moon. 262–190 BC).Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents see Archimedes' use of infinitesimals. The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives are frequently used to find the maxima and minima of a function. ![]() In operations research, derivatives determine the most efficient ways to transport materials and design factories. The reaction rate of a chemical reaction is a derivative. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. ![]() For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.ĭifferentiation has applications to nearly all quantitative disciplines. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. The process of finding a derivative is called differentiation. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. ![]() The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. It is one of the two traditional divisions of calculus, the other being integral calculus. Application Of Differential Calculus Application Of Differential Calculus In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.
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