Differential calculus, a field in mathematics, is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the behavior of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In general, the derivative of a function at a point determines the best linear approximation to the function at that point.
The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.
Differentiation has applications to all quantitative disciplines. 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. By applying game theory, differentiation can provide best strategies for competing corporations.
Derivatives are frequently 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 and abstract algebra.
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