Chapter 2: Differentiation
Section 2.4: The Chain Rule
Polynomials and rational functions can be differentiated with the Differentiation rules in Table 2.3.1. If these or other functions appear in a composition, the Chain rule is required. The Chain rule is the essential element in this section. Its use is illustrated by means of the six examples listed in Table 2.4.1. The function differentiated by the Chain rule in each example is given in Table 2.4.1. The details of the application of the Chain rule appear in Examples 2.4.1-6.
Table 2.4.1 The examples in Chapter 2.4
Composition of Functions
The function Fx=x2+13 is the composition of the functions f⁡x⁢=⁢x3 and g⁡x⁢=⁢x2+1, so that f⁡x⁢=⁢f⁡g⁡x. Thus, F⁡x is called a composite function.
The number F⁡2 is computed by substituting x⁢=⁢2 into x2+1 to get 5, then cubing 5 to get 125. 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. Although g⁡x⁢=⁢x2+1 can itself be thought of as a composite function, in the context of differentiation that will usually not be necessary. What is crucial for differentiation is the correct recognition of the outer and inner functions.
Using purely functional notation, the composition of the functions f and g would be written F=f⁢∘⁢g. Using this notation, the value of the composite function F at x is expressed by the notation F⁡(x)=(f⁢∘⁢g)⁡(x). The notation F⁡(x)=f⁡(g⁡(x)) generally proves to be simpler to use. (The functional notation f∘g looks much like the word "fog" and often causes it.)
Additional examples of composite functions appear in Table 2.4.2.
Table 2.4.2 Composite functions F⁡x⁢=⁢f⁡g⁡x and their component parts
(The word composition has but one pronunciation; the word composite, two. It is pronounced com-POS'-it in the USA, and COMP'-o-zit in other parts of the English-speaking world.)
The Chain Rule
Differentiation of a composite function generally requires the chain rule. 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.
A verbal rendition of the Chain rule, using the terms outer and inner functions:
The derivative of the outer function, evaluated at the inner function, times the derivative of the inner function.
A verbal rendition of the Chain rule, using the terms "wrapper" for the outer function and "stuff inside" for the inner function.
The derivative of the wrapper evaluated at the "stuff inside" times the derivative of the stuff inside.
The phrase "the derivative of the stuff inside" is probably the most important mnemonic that can be attached to the study of the chain rule.
Theorem 2.4.1 is a formal statement of the chain rule.
f and g are differentiable functions
f′g⁡x and g′x both exist
Fx=fg⁡x, the composition of f with g, is differentiable at x
One way to think of the Chain rule is to think of composition as a gear. For example, if a chain is being pulled through a 53-tooth gear and if the gear is being turned at a rate of 100 revolutions per minute then the chain passes through the gear at a rate of (53 links/revolution)(100 revolutions/min) = 5300 links/min. Note how the composite rate is the product of the individual rates.
The Inverse-Function Rule
The function g is the (functional) inverse of the function f if the compositions fgx and gfx both simplify to x. The typical notation for the function inverse to f is f−1, so the compositions that must hold are ff−1x=x=f−1fx.
Let g be the functional inverse of f. The derivative of g is given by the Inverse-Function rule.
Thus, the derivatives of inverse functions are related as reciprocals, but the derivative of the reciprocated function must be evaluated at the inverse function. This rule follows easily from the Chain rule and the identity fgx=x that holds when f and g are inverse functions. The Chain rule gives f′gx⋅g′x=1, from which the reciprocal rule follows by division.
Slopes of Curves Defined Parametrically
If a curve C is defined parametrically by x=xt,y=yt, the slope along the curve is given by
where y^x=ytx. Unfortunately, nearly all texts fail to distinguish y^ from y, even though these are completely different functions. When this notational omission is made, the formula for the derivative of y^ assumes the perplexing form
The most appropriate form for this formula would be the very explicit
which follows from an application of the Chain and Inverse-Function rules for derivatives.
The prime symbol used for differentiation represents an operator. By default, Maple interprets the prime to be the operator ddx. If the differentiation variable is other than x, then one way to have Maple re-interpret the prime is to include the independent variable explicitly. So, if using the prime to obtain the ratio of derivatives, include the appropriate independent variable as an explicit argument.
As a final caution, note that shortening the differentiation rule to something like y′=y′/x′ would befuddle even the most astute reader.
Table 2.4.3 summarizes the key ideas in Section 2.4.
Composite functions are differentiated by the Chain rule.
The second derivative of a composite function is obtained by differentiating the first derivative.
Higher-order derivatives of a composite function are obtained by repeated differentiation.
The kth-derivative of a polynomial of degree n is a polynomial of degree n−k, provided n≥k.
For rational functions, repeated application of the Quotient rule generally results in rational functions with denominators of increasing degree.
The inverse function rule: The derivative of the inverse function is the reciprocal of the derivative, properly evaluated. Thus, if gx=f−1x, then g′x=1/f′gx.
The slope along a curve defined parametrically: ddxy^x=y′tx′tx=a|f(x)t=tx.
Table 2.4.3 Section summary
Use the Chain rule to differentiate F⁡x=x4+3⁢x2+12.
Use the Chain rule to differentiate F⁡x=x4+120.
Use the Chain rule to obtain the second derivative of Fx=fgx.
Obtain the second derivative of Fx=x4+15.
Use the Chain rule to obtain the derivative of Fx=x−πx2−47.
Use the Chain rule to obtain the derivative of the composite function Fx=fghx.
Use the Inverse-Function rule to obtain the derivative of gx=x from the derivative of fx=x2.
If C is the ellipse defined parametrically by x=uθ=5 cosθ, y=vθ=3 sinθ, 0≤θ≤2 π, graph C and obtain the slope of the line tangent to C at the point corresponding to θ=π/6.
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