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The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse operation. This mirrors the conventional way the related theorems are presented in modern basic ...
The inverse chain rule method (a special case of integration by substitution) Integration by parts (to integrate products of functions) Inverse function integration (a formula that expresses the antiderivative of the inverse f −1 of an invertible and continuous function f, in terms of f −1 and the antiderivative of f).
Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component of (those constants need not be equal). This observation implies that if a function g : U → C {\displaystyle g:U\to \mathbb {C} } has an antiderivative, then that antiderivative is unique up to addition of a function which ...
Taylor's theorem; Rules and identities; Sum; Product; Chain; Power; ... of non-zero order k is ... A tensor form of a vector integral theorem may be obtained by ...
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n , one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0 .
Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed ...
The q-derivative of a function is defined by the formula () = () (). For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative, thus the q -derivative may be viewed as its q-deformation .