a, r

Figure 11.3 Some or the infinite number of palhs to approach a point inK2

Figure 11.3 Some or the infinite number of palhs to approach a point inK2

illustrated in figure 11.3, where the paths marked P\. P2 P$ are clearly just a lew examples.

In section 4.1 we also considered an alternative definition of continuity. In definition 4.4. which states that a function is continuous at a point.v = a if, within a small neighborhood of this point (i.e., for points close to x = a), the function values J'(x) are close to the value fia). This definition of continuity extends to functions delined on R" in a straightforward manner. For .v € R, closeness is determined by the (absolute) distance |.v — ci\. To extend the concept of continuity to functions on x e R", we can simply use the Euclidean distance between the points, ||x - a||. introduced in chapters 2 and 10. For example, in R- we have

||x — a|| = ||(xi, - Utr,¿>2)11 = v/U - ft)3 + ix2 - a2)1

We do not study this definition of continuity further here, though question 12 at the end of this section illustrates how to use this definition for functions on R".

Extending the idea of the derivative for functions of one variable to functions delined on K" is more straightforward than is continuity. Recall from chapter 5 that the derivative of a function y = f(x) with domain x e R is the rate at which v changes as jr changes (Ay/A.v) as we let the change in a become arbitrarily small (A.* —►()). If y depends on more than one variable, as in y = fixt,x2) we can define the rate at which y changes with respect to changes in each of the variables .V| and .v2. taken separately, in the same way. For example, we can lind the ratio A_v/ as A.\ | —* 0 while holding x2 constant. The result of this operation is called the partial derivative of the function y - fix|, x2) with respect to the variable ,\ |. and it is written

Ojc i tijc L


• a"2) .. J'(xi -I- A.t), x2) - f U,, jc2) --= lim -----——----—

The partial derivative of the function f(x\,x2) with respect to the variable x2 is written

---—— or — or /i(x i.-tj) or simply fi ox2 3x2

where a/a,. x2) ,. /(.v,. x2 + Axn) - t'ix\.x2) - = lim -—------

The reason for calling these expressions partial derivatives and using the notation d" is that we are changing only one of x\ or x2 at a lime, even though y depends on both of these variables. It is important to remember that since y is a function nf both ati and x2, the ratio Ay/Axi will in general depend on the level of a> Similarly the ratio Ay/Ax? will depend on the level of n. The notation /i(xi, \V> und /2(.vi , ,v;>) reminds us that the rale of change of y with respect to | or x2 is itself a function which in general depends on the values of both A| and x2.

The idea of the derivative being the slope of the tangent to ihe curve at some point in the one-variable case carries over to the case where .\ 6 However, one must take care in drawing and interpreting the relevant diagram.

Notice in figure 11.4 (a) that with x;> fixed, there are now only two directions, lather than an arbitrary number of directions, from which to approach any given point. This being the case, the partial derivative 3 v/3Jtj behaves just like the derivative of a function of one variable since only one variable, xi, is changing. This similarity with the one-variable case explains why die process of partial differentiation is a straightforward extension of Ihe derivative for functions of one variable.

Figure 11.4 The partial derivatives for a function on K.J

The idea of the partial derivative generalizes readily to functions of n variables.

Definition 11.1

The partial derivative of a function y - f{x\. x;,.. ..v„) with respect to the variable x, is

The notations Hy/dx; or /; (x) or simply J) are used interchangeably. Notice that in defining the partial derivative /}<x) all other variables, Xj, j ^ i. arc held constant.

As in the case of the derivative of a function of one variable, we can use definition 11.1 to compute the partial derivatives of a specific function. The following example illustrates.

Example 11.1 Derive and interpret the partial derivatives of the revenue function for a multi-product. competitive firm.


Suppose that we let .tj and x2 represent the quantities of two products sold by a competitive firm with p\ and p2 being the priceof each, respectively. Total revenue for the firm is R(x\. x2) = p\X\ 4- p2x2. According to definition 111. the panial derivative, x2)/$x\ is

The derivation of R2(x¡,x2) is left as one of the exercises at die end of this section. ■

The result R,(xi,x2) = p\ accords with simple intuition. The amount by which revenue increases as one mure unit of good 1 is sold with the output of the other good left unchanged is simply the price of good I. Notice that this derivative is just a constant. This result is the same as for the case of a linear function of one variable. The independence of R\ with respect to the value of A| ot a"i carries over for any linear function. It is of course not true lor functions in general, as the following example illustrates.

Example 11.2 In this example the derivative by/ih] of the function y = xjx2 depends on the value of both .v, and v?. According to definition I I.I. the partial derivative, fl/(,T|, Xi)/BX\, is

Rather than derive them from first principles (i.e.. by using (he definition of the derivative), we can use rules of differentiation to find partial derivatives just as we did for functions of one variable in chapter 5. Since we hold all variables except .v, fixed when finding itf/'d.x,. we can explicitly treat all parts of the function fix) that do not depend on x, as a constant, c, and then use the rules of differentiation for functions of one variable. For the function of example 11.1. R(x\. xz) — p\X\ + pzxz\ this means setting p2xz = c, where c is some constant. Then noting that

Similarly, for the function of example 11.2, y = fix\. xz) = x2x2. the variable xz is held fixed when computing 'df/'dx\. If we explicitly set X: - c. c a constant, then the function becomes

and so

tix i which, upon substituting back for c = x2, gives the result

Definition 11.2

Now let us see how the procedure above carries over to functions of any number of variables. If we have y = f'.x\ .x2 *,,). then to find the derivative of / with respect to one of the variables, jr,. we factor out ail pans of the expression that are not dependent on .v, and treat these as constant values. For example, given the function y =

we find the partial derivative 0y/3xi by first setting

It follows from the rules of differentiation given in chapter 5 that

which, upon substituting back for c = gives the result that

After a little practice you don't need to make these substitutions explicitly.

We can see from these examples that the partial derivative 'df/Bx, will in general depend on the values of all variables, Xj, even for /' ^ i. There is a class of functions, however, that has the property that the partial derivative with respect to v, is independent of all the other variables xj, j / <. We define this class by

A function y = /(jci . x2 x„) which can be written in the form fix,, xi x„) = g'(x J ) + r<.vj) + ••• + g"(xn)

0 0

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