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Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation . Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point.
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. Differential calculus and integral calculus are connected by the fundamental theorem of calculus. This states that differentiation is the reverse process to integration.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. [1]: 26ff A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.
Mathematical analysis formally developed in the 17th century during the Scientific Revolution, [3] but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics.
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
Marston Morse applied calculus of variations in what is now called Morse theory. [6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. [6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations. [7] [8] [9] [c]
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...