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 2kπ ≤ x ≥ (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), …, u_k–1 = Φ_k–1(u_k), u_k = Φ_k(x), then