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Generalized power rule. The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g , wherever both sides are well defined. Special cases. If , then when a is any non-zero real number and x is positive. The reciprocal rule may be derived as the special case where .
Miscellanea. v. t. e. In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as or in Leibniz's notation as. The rule may be extended or generalized to products of three or more ...
t. e. In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x ...
Red, green and blue arrows represent multiplication by i, j, and k, respectively. Multiplication by negative numbers is omitted for clarity. Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±j, ±k} forms a group under multiplication.
Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain ).
t. e. In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. [1] The set X is called the domain of the function [2] and the set Y is called the codomain of the function. [3] Functions were originally the idealization of how a varying quantity depends on another quantity.
The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition. [ 1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree is the symmetric group on the set .
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