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In 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 study of the area beneath a curve; the primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, 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 called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point 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.

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, the derivative of velocity with respect to time is acceleration; the derivative of the momentum of a body with respect to time 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 used to find the maxima and minima of a function. Equations involving derivatives are called 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, abstract algebra. Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y.

This relationship can be written. If f is the equation for a straight line there are two real numbers m and b such that y = mx + b. In this "slope-intercept form", the term m is called the slope and can be determined from the formula: m = change in y change in x = Δ y Δ x, where the symbol Δ is an abbreviation for "change in", it follows that Δy = m Δx. A general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a; this is denoted f ′ in Lagrange's notation or dy/dx|x = a in Leibniz's notation. Since the derivative is the slope of the linear approximation to f at the point a, the derivative determines the best linear approximation, or linearization, of f near the point a. If every point a in the domain of f has a derivative, there is a function that sends every point a to the derivative of f at a. For example, if f = x2 the derivative function f ′ = dy/dx = 2x. A related notion is the differential of a function.

When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, denoted ∂y/∂x; the linearization of f in all directions at once is called the total derivative. The concept of a derivative in the sense of a tangent line is a old one, familiar to Greek geometers such as Euclid and Apollonius of Perga. Archimedes introduced the use of infinitesimals, although these were used to study areas and volumes rather than derivatives and tangents; the use of infinitesimals to study rates of change can be found in Indian mathematics as early as 500 AD, when the astronomer and mathematician Aryabhata used infinitesimals to study the orbit of the Moon.

The use of infinitesimals to compute rates of change was developed by Bhāskara II. The Islamic mathematician, Sharaf al-Dīn al-Tūsī, in his Treatise on Equations, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials, he proved, for example, that the maximum of the cubic ax2 – x3 occurs when x = 2a/3, concluded therefrom that the equation ax2 — x3 = c has one positive solution when c = 4a3/27, two positive solutions whenever 0 < c < 4a3/27. The historian of science, Roshdi Rashed, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known; the modern develo

Adolph Strasser, born in the Austro-Hungarian empire, was an American trade union organizer. Strasser is best remembered as a founder of the United Cigarmakers Union and the American Federation of Labor. Strasser was additionally the president of the Cigar Makers' International Union for a period of 14 years, heading the union during the period in which it introduced its successful union label and gained substantial organizational strength. Adolph Strasser was born in the Austro-Hungarian empire in part of today's Hungary in 1843, he was a native speaker of German. Strasser emigrated to the United States in 1871 or 1872. After his arrival in America, Strasser worked at the craft of cigar making, taking up residence and employment in New York City. In his posthumous memoirs fellow cigarmaker Samuel Gompers recalled his impressions of Strasser from the time he met him in 1872: "Strasser was a man of extraordinary mentality, he came to America some months before, traveling through the South before settling in New York City.