Rule 3 Derivative of a Power Function fx xn

For example, the derivative of the function f(x) = ,r: is fix) = 2x, as we denved1 from the definition of the derivalive in section 5.2. The following list of examples illustrates various types of results using this rule.

For fix) = x3/2,x > 0. we get f'(x) = 0/2)x^2'^ = Q/2)xt/2. The function fix) = x~2 has derivative fix) = —2x ~3. The derivative of fix) = .v". -v # 0. is fix) = Ox"'1 = 0/x = 0 for .v # 0. Since for x / 0, x° = 1. this is clearly the correct result. (Recall that 0" is not defined.)

Example 5,6 Total and Marginal Product of Labor

Consider the production function y = L", a > 0 which relates the level of input labor. L, to output, y. This function is often called the total product of labor, TP(7j. The marginal product of labor is then MP(L) = dy/dL = aLa~[. If the parameter value for a is greater than I. then the marginal product of labor is increasing in L. For example, for y = L-'2 we get \1P(L) = (3/2)Z-l/2. If a = I, the function is simply the linear function y = L, and the derivative is the constant function dy/dL = 1. If the value for a is less than 1, then the marginal product of labor is decreasing in L. For example, for y - LI/2 we get MP(L) = (1/2)L~i^2 or 1/(2L{/2). Tltese functions are graphed in figures 5.21,5.22, and 5.23.

Figure 5.21 Total product of labor displaying increasing marginal product

Figure 5.21 Total product of labor displaying increasing marginal product

TPiL)

TPiL)

MPiL)

It is reasonable to expect that if the amounts of inputs other than labor are held fixed, then, at least eventually, as more labor is used we obtain smaller increments in output per added unit of labor. This presumption is called diminishing marginal productivity of a variable input. The only time this is satisfied for the functional form y — L" is when a < i, as the above examples illustrate. As a result we often see the restriction a < 1 imposed when the example of y = L" is used. We investigate this issue further in section 5.5. ■

As you will see throughout the chapters on optimization and multivariate calculus, power functions are frequently used to illustrate properties of various ' economic concepts such as production functions and utility functions. For example, the one variable case considered above, y - L". generalizes to the important class of examples referred to as Cobb-Douglas production or utility functions. It is very important, therefore, to understand the relationship between the size of« and the change in the value of the marginal product of labor as L increases.

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