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This states that differentiation is the reverse process to integration. Differentiation has applications in nearly 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 the velocity with respect to time is acceleration .
Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called differentiation. Given a function defined at several points of the real line, the difference quotient at that point is a way of encoding the small-scale (i ...
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly denoted , is the operator that maps a function f to the function [] defined by [] = (+) ().
Thus, differentiation is the process of distinguishing the differences of a product or offering from others, to make it more attractive to a particular target market. [3] Although research in a niche market may result in changing a product in order to improve differentiation, the changes themselves are not differentiation. Marketing or product ...
The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified with the calculus of finite differences in time scale calculus. [53] The arithmetic derivative involves the function that is defined for the integers by the prime factorization. This is an analogy with the product rule. [54]
Integrating this relationship gives = ′ (()) +.This is only useful if the integral exists. In particular we need ′ to be non-zero across the range of integration. It follows that a function that has a continuous derivative has an inverse in a neighbourhood of every point where the derivative is non-zero.
Isaac Newton's notation for differentiation (also called the dot notation, fluxions, or sometimes, crudely, the flyspeck notation [12] for differentiation) places a dot over the dependent variable. That is, if y is a function of t, then the derivative of y with respect to t is