The derivative of the wrapper evaluated at the "stuff inside" times the derivative of the stuff inside. The derivative of the outer function, evaluated at the inner function, times the derivative of the inner function.Ī verbal rendition of the Chain rule, using the terms "wrapper" for the outer function and "stuff inside" for the inner function. Of all the rules of differentiation, the chain rule is the most important, more for theoretical reasons than for the immediate need to evaluate derivatives of specific functions.į x = f g x ⇒ F ′ x = f ′ g x g ′ xĪ verbal rendition of the Chain rule, using the terms outer and inner functions: It is pronounced com-POS'-it in the USA, and COMP'-o-zit in other parts of the English-speaking world.)ĭifferentiation of a composite function generally requires the chain rule. (The word composition has but one pronunciation the word composite, two. Table 2.4.2 Composite functions F x = f g x and their component parts (The functional notation f ∘ g looks much like the word "fog" and often causes it.)Īdditional examples of composite functions appear in Table 2.4.2.
The notation F ( x ) = f ( g ( x ) ) generally proves to be simpler to use. Using this notation, the value of the composite function F at x is expressed by the notation F ( x ) = ( f ∘ g ) ( x ). Using purely functional notation, the composition of the functions f and g would be written F = f ∘ g. What is crucial for differentiation is the correct recognition of the outer and inner functions. Although g x = x 2 + 1 can itself be thought of as a composite function, in the context of differentiation that will usually not be necessary. The first function that must be evaluated will be called the "inner" function, while the second function that gets applied afterwards will be called the "outer" function. The number F 2 is computed by substituting x = 2 into x 2 + 1 to get 5, then cubing 5 to get 125.
Thus, F x is called a composite function. The function F x = x 2 + 1 3 is the composition of the functions f x = x 3 and g x = x 2 + 1, so that f x = f g x. The details of the application of the Chain rule appear in Examples 2.4.1-6.į x = x 4 + 3 x 2 + 1 2 The function differentiated by the Chain rule in each example is given in Table 2.4.1. Its use is illustrated by means of the six examples listed in Table 2.4.1. The Chain rule is the essential element in this section. If these or other functions appear in a composition, the Chain rule is required. Polynomials and rational functions can be differentiated with the Differentiation rules in Table 2.3.1.
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