# Composite Function

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## composite function

[kəm′päz·ət ′fəŋk·shən]
(mathematics)
A function of one or more independent variables that are themselves functions of one or more other independent variables.

## Composite Function

a function of a function. Let us suppose that the variable y is a function of u, that is, y = f(u), and that u is in turn a function of x, that is, u = Φ(x). Then y is a composite function of u and x—that is, y = f[Φ(x)]—defined for all x such that Φ(x) is in the domain of f(u). In this case, y is said to be a composite function of the variable x and the variable u, which is sometimes called the intermediate variable. For example, if y = u2 and u = sin x, then y = sin2x for all values of x. If, however, and u = sin x, then , which, if we are restricted to real values of the function, is defined only for all x such that sin x ≥ 0—that is, for 2kπx ≥ (2k + 1)π, where k = 0, ± 1, ± 2,….

The derivative of a composite function is equal to the product of the derivative of the function with respect to the intermediate variable and the derivative of the intermediate variable with respect to the independent variable. This rule, called the chain rule, extends to composite functions with two, three, or more intermediate variables. Thus, if y = f(u1, u1 = Φ(u2), …, uk–1 = Φk–1(uk), uk = Φk(x), then Mentioned in ?
References in periodicals archive ?
In this paper, the method of total derivative of composite functions of several variables and the structure is used to convert system, expressed as state space form or as state space form, into cascaded integral system suitable for ADRC application.
The Structure Method of the Total Derivative of Several Variables Composite Functions
The proposition is the extension of the theorem from  and it shows how the partial derivative of the total delta derivative of the composite function with a vector argument can be expressed through the total delta derivative of the partial derivative of the composite function.
Chen: An explicit formula of Hessian determinants of composite functions and its applications, Kragujevac J.
It was noted that the prerequisite knowledge of composite function is another significant notion for applying the chain rule by raising awareness of the relation among various cases.
I also introduce the idea of composite functions, that is, using the output of one machine as the input for a second machine.
The main tool for proving the theorem (commutation rule) is Mishkov's theorem  which provides the explicit formula for the nth derivative of a composite function with a vector argument.
We continue our considerations by pointing out a result on the subdifferential of composite functions.
But at present, none had studied the mean value properties of the composite function S([P.
The 14 papers include discussions of the reducibility of Schlesinger isomonodromic families, the Cauchy problem in a half-plane for a hyperbolic equation with constant coefficients, zeros of some composite functions, periodic entropy sub-solutions and super-solutions to a degenerate elliptic-hyperbolic equation, and Mellin convolution operators in variable exponent Lebesgue spaces.
The Japanese patent #2008109628B has 21 claims which include the "Howard Cascade" for problem distribution and answer agglomeration, serial and parallel I/O, and run-time node allocation/de-allocation (used in the Blue Cheetah - Coalition product) as well as composite functions and algorithm/kernel definitions (used in Blue Cheetah - Cub.
The patent's Composite Functions Development model offers a new programming technique for parallel systems.

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