## Rules Of Differentiation 185

Example 5.10

Figure 5.25 Marginal revenue foi u monopolist facing the inverse demand function p — 40 - 2q

Notice the change in sign to positive for the second term. This simply recognizes that in the expression for AR/Aq. we treat the reduction in price as a positive value, while in the expression lim.^-^o AR/Aq. die term dp/dq is the slope of the inverse demand function which is itself negative (i.e.. —Ap/Aq is the slope of the inverse demand function, dp/dq, as Aq —*■ 0) Thus the second term in the expression above for MR is indeed negative.

An example with a specific functional form may illuminate further. Suppose that the demand function is linear and, in its inverse form.

The monopolist's total revenue function becomes

JR(q) = p(q)q = [40-2q]q Using the product rule, the marginal revenue function is

MR(a) = ————————<7 + —[40 - 2q] = \-2\q -f 1 [40 — 2q\ = 40-4q dq dq

As an example ol the fact that MR < p. notice that for output level q = 5, the price charged is p = 30 but MR(4) = 20, which is less than price. This is illustrated in figure 5.25. ■

Rule 8 Derivative of the Quotient of Two Functions tfh(x) = f(x)/g(x) and g(x) # 0, then

Example 5.11 Find the derivative of

x - 5x Solution

We can find the derivative by letting fix) = lO.r2 + 3x4 + 5 and ,ç(.v) = x2 - 5.v and then using the formula to get fix ) = 20.t + l 2x3 and g'ix ) = 2.r - 5

and so

(20.* + 12jc3)(a-2 - 5a-) - (2a- - 5)( I0x2 + 3a4 + 5) h t*) =-——-

Relation between Average and Marginal Values of a Function

Given a function fix) (which may be a cost function, total product function, etc.) its average value function is

Using the quotient rule, it follows that

X' x x J x where fix) is the marginal value function. This tells us the following:

(i) When fix) < A(.r), then A'ix) < 0; that is, the average value function is falling.

(ii) When fix) = A(.r), then A'ix) = 0: that is, the average value function is horizontal or is at a point of horizontal tangency.

(iii) When f'(x)>A(x), then A'ix) > 0; that is. the average value function ii rising.

This relationship is illustrated for an average cost function, ACiy), and marginal cost function. MCiy). in figure 5.26. Note that when the cost of producing

an exiru unit of output, which is the marginal cost, is below the average cost, such as at point y = yj, the average cost of production is falling in y. When the marginal cost is equal to the average cost, such as at point y = vj, the average cost of production is not changing. When the marginal cost exceeds average cost, such as at point y - V3. the average cost of production is rising.

Example 5.12 For the total cost function

show that

(i) MC is less than AC where AC is falling

(ii) MC = AC at the point where the AC curve is horizontal (Hi) MC exceeds AC where AC is rising.

Solution