Rules of Differentiation and Their Use in Comparative Statics

The central problem of comparative-static analysis, that of finding a rate of change, can be identified with the problem of finding the derivative of some function y = provided only an infinitesimal change in is being considered. Even though the derivative dv/dx is defined as the limit of the difference quotient q = g( v) as i; —► 0, it is by no means necessary to undertake the process of limit-taking each time the derivative of a function is sought, for there exist various rules of differentiation (derivation) that will enable us to obtain the desired derivatives directly. Instead of going into comparative-static models immediately, therefore, let us begin by learning some rules of differentiation.

7.1 Rules of Differentiation for a Function of One Variable

First, let us discuss three rules that apply respectively, to the following types of function of a single independent variable: v = k (constant function} and y = xn and y — cxn (power functions). All these have smooth, continuous graphs and are therefore differentiate everywhere.

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