Application Of Differential CalculusApplication Of Differential CalculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the
rates at which quantities change. It is one of the two traditional divisions of calculus, the other
being integral calculus.
The primary objects of study in differential calculus are the derivative of a function, related
notions such as the differential, and their applications.
The derivative of a function at a chosen input value describes the rate of change of the
function near that input value.
The process of finding a derivative is cal ed differentiation. Geometrically, the derivative at a
point equals the slope of the tangent line to the graph of the function at that point.
For a real-valued function of a single real variable, the derivative of a function at a point
general y determines the best linear approximation to the function at that point.
Differential calculus and integral calculus are connected by the fundamental theorem of
calculus, which states that differentiation is the reverse process to integration.
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Differentiation has applications to nearly all quantitative disciplines. 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.
Newton's second law of motion states that the derivative of the momentum of a body equals
the force applied to the body.
The reaction rate of a chemical reaction is a derivative. In operations research, derivatives
determine the most efficient ways to transport materials and design factories.
Derivatives are frequently used to find the maxima and minima of a function. Equations
involving derivatives are cal ed differential equations and are fundamental in describing
natural phenomena.
Derivatives and their generalizations appear in many fields of mathematics, such as complex
analysis, functional analysis, differential geometry, measure theory and abstract algebra.
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. 300 BC), Archimedes (c. 287-212 BC) and Apol onius of Perga
(c. 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 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.
The use of infinitesimals to compute rates of change was developed significantly by Bhskara
II (1114-1185); indeed, it has been argued[3] that many of the key notions of differential
calculus can be found in his work, such as "Rolle's theorem".
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The Persian mathematician, Sharaf al-Dn al-Ts (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 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 key insight, however, that earned them this credit, was the
fundamental theorem of calculus relating differentiation and integration: this rendered obsolete
most previous methods for computing areas and volumes, which had not been significantly
extended since the time of Ibn al-Haytham (Alhazen).
For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by
mathematicians such as Isaac Barrow (1630-1677), Rene Descartes (1596-1650), Christiaan
Huygens (1629-1695), Blaise Pascal (1623-1662) and John Wal is (1616-1703). Isaac
Barrow is general y given credit for the early development of the derivative.[10] Nevertheless,
Newton and Leibniz remain key figures in the history of differentiation, not least because
Newton was the first to apply differentiation to theoretical physics, while Leibniz systematical y
developed much of the notation still used today.
Since the 17th century many mathematicians have contributed to the theory of differentiation.
In the 19th century, calculus was put on a much more rigorous footing by mathematicians
such as Augustin Louis Cauchy (1789-1857), Bernhard Riemann (1826-1866), and Karl
Weierstrass (1815-1897). It was also during this period that the differentiation was
generalized to Euclidean space and the complex plane.
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