chapter7Notes - Chapter 7 The Chain Rule Related Rates and...

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Chapter 7 The Chain Rule, Related Rates, and Implicit Differentiation 7.1 Function composition x u y f g Figure 7.1: Shown in the diagram above is an example of function composition: An independent variable, x , is used to evaluate a function, and the result, u = f ( x ) then acts as an input to a second function, g . The result, y = g ( u ) = g ( f ( x )) can be related to the original variable, and we are interested in understanding how changes in that original variable affect the final outcome: That is, we want to know how y changes when we change x . The chain rule will apply to this situation. 7.2 The chain rule The chain rule of differentiation helps to calculate the result of this chain of effects. Basically, this rule states that the change in y with respect to x is a product of two rates of change: (1) the rate of change of y with respect to its immediate input u , and (2) the rate of change of u with respect to its input, x . If y = g ( u ) and u = f ( x ) are both differentiable functions (meaning that their derivatives exist everywhere), and we consider the composite function y = g ( f ( x )) then the chain rule says that dy dx = dy du du dx It is common to use the notation d/dx as shown here when stating the chain rule, simply because this notation helps to remember the rule. Although the derivative is not merely a quotient, we can recall that it is arrived at from a quotient through a process of shrinking an interval. If we write Δ y Δ x = Δ y Δ u Δ u Δ x v.2005.1 - September 4, 2009 1
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Math 102 Notes Chapter 7 then it is apparent that the “cancellation” of terms Δ u in numerator and denominator lead to the correct fraction on the left. The proof of the chain rule (optional) uses this essential idea, but care is taken to ensure that the quantity Δ u is nonzero, to avoid the embarrassment of dealing with the nonsensical ratio 0 / 0. The most important aspect of the chain rule to students of this course is an understanding of why it is needed, and how to use it in practical examples. The following intuitive examples may help to motivate why the chain rule is based on a product of two rates of change. Later in this chapter, we discuss examples of applications of this rule. 7.2.1 Example 1: A species of fish is sensitive to pollutants in its lake. As humans settle and populate the area adjoining the lake, one may see a decline in the population of these fish due to increased levels of pollution. The rate of decline of the fish would depend on the rate of change in the human population around the lake, and the rate of change in the pollution created by each person. If either of these factors increases, one would expect an increase in the effect on the fish population and their possible extinction. The chain rule says that the net effect is a product of the two interdependent rates. To be more specific, we could think of time t in years, x = f ( t ) as the number of people living at the lake in year t , and p = g ( x ) as the pollution created by x people. Then the rate of change of the pollution p
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