The derivative of a function y - /(,v) at some point in the domain of the function is simply the slope of the tangent line. The notation used to depict the derivative at a point, x, in the domain of the function f(x) varies but is usually written either as dy/dx or f'(x). Since, its we saw in the previous section, the slope ol the tangent line generally depends on the value of the variable .v. then so does the value of the derivative. Therefore, the derivative function. /', is theTunction which indicates the value of ihe derivative of the function at each point of the domain of /. Only in the case of a linear function, y = >nx + is the value of the derivative independent of the value off. In this case the derivative is fix) = hi ai every point x in the domain and so the derivative function is /' - m. the constant function. (Note that, as in Chapter 4. the domain of a function is assumed to be: unless otherwise stated.)
The derivative of a function y = /(x) at the point P = Ui. f(xt )) is the slope of the tangent line at that point.
Ax-»0 A.t-'O X2— -ti where Ax = x2 - ,V|. We can also write this as
In the previous section, we showed that the slope of the tangent line for the function y = x2 is 2x and so this is also its derivative; i.e.. f'(x)~2x or dy/dx — 2.V From an intuitive perspective, notice that the dy and dx reflect the idea of changes in y and x. as do Ay and A.v. respectively. In fact, for a specific value of dx we can think of dy/dx as an estimate of the ratio Ay/A.v, and so dy/dx = 2.v can be written as dy = (2x)dx with dy representing an estimate of Ay for the value of dx chosen to be equal to A.v. The expression dy = (2x)dx is known as the differential of the function y = x2. A formal definition for the differential is given below.
If /'(.v") is the derivative of the function y = /(,v) at the point a". then the total differential at a point a:0 is dy - df(x°. dx) = f\xu) dx
Thus, the differential is a function of both .v and dx.
The differential provides u> with a method of estimating the effect of a change in Af amount dx ~ A.v on y, where Ay is the exact change in y while dx is the approximate change in y. Given the definition of the derivative, this is equivalent to using the tangent line of a function to estimate the impaci of a change in a on y. For the function f(x) = X2. the differential is dy = f'(x) dx = 2.v dx To see that this expression represents only an approximation to the true relationship between the change in x and the change in y. refer to figure 5.8. We see that Irom the point P = (2. 4). an increase in x of amount A.v = dx - 2 leads to a change in i of A v = 12. If we use the tangent line at the point P ~ (2. 4) to estimate Lhe change in y resulting from a change in x of 2 units, we find that dy = 8 (i.e., dy = f'(x) dx = (2,v)dx = 4 dx, with dx — 2). In this case, using lhe differential leads to an estimate of Ay that is too low.
In fact, in either the derivative or differential form, the relationship between dx and dy can be thought of as an estimate for the true relationship between changes in x and v (i.e., A.v and Ay). This is clearer if we think of the relationship in the following form:
where i is the approximation error. For the example illustrated in figure 5.8 the error is e = 4. Thus, as an approximation of the true change in y. the formula of equation (5.3) is not very impressive. However, one can see that for smaller changes in x, the expression dy — f'(x) dx offers a belter approximation. For example, beginning again with v = 2 (i.e., P = (2,4)), we find that choosing
Ax = dx = I leads to Ay = 5 and dy = 4 (see figure 5.9). Not only is dy closer to Ay in absolute terms, but the percentage error is reduced from 33% (4/12) to 20% (1/5) when going from Ax = 2 to Ax = 1. Furthermore we can show that equation (5.3) can be made arbitrarily accurate (i.e., the percentage error f/Ay can be made arbitrarily small) by requiring that the change in x be small. Formally this means that limA.t=</i-of/Ay = 0.
Notice that for the example v = x2, using the tangent line or the expression dy — f'(x)dx to approximate the impact of a change in x on y led lo an underestimate. It is of course also possible that the use of the tangent line will lead to an overestimate of the impact of A.v on Ay. Such is clearly the case for the function illustrated in figure 5.10. For the case of a linear function, y = mx + b. the expression dy = f'(x)dx = m dx provides an exact approximation of the impact of a change in x of amount dx or A.v on y (i.e.. dy — m dx and Ay = m Ax). This is illustrated in figure 5.11.
The Total- and Marginal-Cost Functions
A firm's total-cost function, C - C(y), indicates the cost of producing amount of output, y. Thus, given C = C(y), the ratio AC/Ay = (C(y -+ Ay) - C(y))/Ay reflects the (average) rate of change in cost per added unit of output produced. If we take the limit of this ratio as Ay -> 0, wc get the instantaneous rate of change, which is generally referred to as the marginal-cost of production:
Aj -o Ay Ay—o Av and is the derivative of the total-cost function.
Let us begin with the simplest type of example, the case of a linear cost function. In particular, let C = 80_v. This function implies that whatever is the current level of output produced, the cost of producing an extfa unit of output is 80 (i.e., since C'(y) or dC/dy - 80). The differential dC = C'(y)dy becomes dC = 80 dy, indicating that any change in y of amount dy leads to a change in cosi of 80 times dy (e.g., producing dy = 3 more units of output leads to an increase in cost of dC = 80(3) = 240). As indicated earlier in this section (see figure 5.11). in the case of a linear function the differential represents an exact estimate of the relationship between the actual change in C(i.e., AC) and the actual change in y(i.e..Ay). Moreover the fact that the derivative is a constant implies that the marginal cost of production is independent of the existing amount of output being produced. This is illustrated by figure 5.12.
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