Thus, by itself Ay/Ax (without requiring Ax 0) is not equal to dy/dx. If we denote the discrepancy between the two quotients by ¿, wc can write

—r- - -f- = s where 8 0 as Ax 0 [by (8.1)] (8,2) Ax dx

Multiplying (8.2) through by Ax, and rearranging, we have

This equation describes the change my {Ay) that results from a specific—not necessarily small—change in x (Ax) from any starting value of x in the domain of the function

FIGURE 8.1

FIGURE 8.1

to W

y — j\x). But it also suggests that we can, by ignoring the discrepancy term 8 Ax, use the fix) Ax term as an approximation to the true Ay value, where the approximation gets progressively better as Ax gets progressively smaller.

In Fig. S.ltf, when x changes from jcq to + Ax, a movement from point A to point B occurs on the graph ofy = /(a). The true Ay is measured by the distance CB< and the ratio of the two distances CB/AC = Ay/Ax can be read from the slope of line segment AB. But if we draw a tangent linC/LD through point A, and use AD in place of AB to approximate the value of Ay, we obtain distance CD, which leaves distance DB as the discrepancy or error of approximation. Since the slope of AD is f(xn), distance CD is equal to /'(xo) Ax and, by (8.3), distance DB is equal to <5 Ax. Obviously, as Aa decreases, point B would slide along the curve toward point A, thereby reducing the discrepancy and making f'(x) or dy/dx abetter approximation to Ay/Ax.

Focusing on the tangent line AD, and taking the distance CD as an approximation to CIL let us relabel the distances AC and CD by dx andd>\ respectively, as in Fig. 8.1/?. Then dy

— = slope of tangent AD ~ j (x) dx and, after multiplying through by dx, we get dy - f(x) dx (8.4)

The derivative f(x) can then be reinterpreted as the factor of proportionality between the two finite changes dy and dx. Accordingly, given a specific value of dx, we can multiply it

Chapter 8 Comparative-Static Analysis of General-Function Models 181

by /'(*) to get dy as an approximation to A y, with the understanding that the smaller the Ax, the better the approximation. The quantities dx and dy are called the differentials of x and y, respectively.

A few remarks are in order regarding differentials as mathematical entities. First, while dx is an independent variable, dy is a dependent variable. Specifically, dy is a function of x as well as of dx\ It depends on x because a different position for x0 in Fig. 8.1 would mean a different location for points and for its tangent line; it depends on dx because a different magnitude of dx would mean a different position for point C as well as a different distance CD. Second, if dx = 0, then dy = 0, because point B would in that case coincide with point A. But if dx ^ 0, then it is possible to divide dy by dx to get fix), just as we can multiply dx by fix) to get dy. Third, the differential dy can be expressed only in terms of some other differential{s)—here, dx, This is because our context calls for the coupling of a dependent change dy with an independent change dx. While it makes sense to write dy = f{x) dx, it is not meaningful to chop away the dx term on the right and write dy = f(x). The coupling of the two changes is effected through the derivative f'(x), which may be viewed as a "converter" that serves to translate a given change dx into a counterpart change dy.

The process of finding the differential dy from a given function y = fix) is called differentiation. Recall that we have been using this term as a synonym for derivation, without having given an adequate explanation. In light of our interpretation of a derivative as a quotient of two differentials, however, the rationale of the term becomes self-evident, it is still somewhat ambiguous, though, to use the single term "differentiation" to refer to the process of finding the differential dy as well as to that of finding the derivative dyfdx. To avoid confusion, the usual practice is to qualify the word differentiation with the phrase "with respect to x" when we take the derivative dyfdx.

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