The last section gave several economic examples of functions involving many variables.
Accordingly, we need to extend the concept of partial derivative to functions with more than two variables.
P A R T I A L D E R I V A T I V E S I N N V A R I A B L E S
Ifz=f (x)=f (x1, x2, . . . , xn), then∂f/∂xi, fori =1,2, . . . , n, means the partial derivative of f (x1, x2, . . . , xn)w.r.t. xi when all the other variablesxj (j =i) are held constant.
(1)
S E C T I O N 1 1 . 6 / P A R T I A L D E R I V A T I V E S W I T H M O R E V A R I A B L E S 401 So provided they all exist, there arenpartial derivatives of first order, one for each variable xi,i =1,. . .,n. Other notation used for the first-order partials ofz=f (x1, x2, . . . , xn) includes
∂f
∂xi = ∂z
∂xi =∂z/∂xi =zi =fi(x1, x2, . . . , xn)
E X A M P L E 1 Find the three first-order partials off (x1, x2, x3)=5x12+x1x23−x22x32+x33. Solution: We find that
f1=10x1+x23, f2=3x1x22−2x2x32, f3= −2x22x3+3x32 As in (11.2.6) we have the following rough approximation:
The partial derivative∂z/∂xiis approximately equal to the per unit change in z=f (x1, x2, . . . , xn)
caused by an increase inxi, while holding constant all the otherxj(j =i).
In symbols, forhsmall one has fi(x1, . . . , xn)≈
f (x1, . . . , xi−1, xi+h, xi+1, . . . , xn)−f (x1, . . . , xi−1, xi, xi+1, . . . , xn) h
(2)
For each of thenfirst-order partials off, we havensecond-order partials:
∂
∂xj ∂f
∂xi
= ∂2f
∂xj∂xi =zij
Here bothiandj may take any value 1,2, . . . , n, so altogether there aren2 second-order partials.
It is usual to display these second-order partials in ann×nsquare array as follows
f(x)=
⎛
⎜⎜
⎜⎜
⎜⎝
f11(x) f12(x) . . . f1n(x) f21(x) f22(x) . . . f2n(x)
... ... . .. ... fn1(x) fn2(x) . . . fnn(x)
⎞
⎟⎟
⎟⎟
⎟⎠
(TheHessian) (3)
Such rectangular arrays of numbers (or symbols) are called matrices, and (3) is called theHessian matrixoff atx =(x1, x2, . . . , xn). See Chapter 15 for more discussion of matrices in general.
Thensecond-order partial derivativesfiifound by differentiating twice w.r.t. the same variable are calleddirect second-order partials; the others,fijwherei =j, aremixedor crosspartials.
E X A M P L E 2 Find the Hessian matrix off (x1, x2, x3)=5x12+x1x23−x22x32+x33. (See Example 1.) Solution: We differentiate the first-order partials found in Example 1. The resulting Hessian
is ⎛
⎝f11 f12 f13 f21 f22 f23 f31 f32 f33
⎞
⎠=
⎛
⎝ 10 3x22 0 3x22 6x1x2−2x32 −4x2x3
0 −4x2x3 −2x22+6x3
⎞
⎠
Young’s Theorem
Ifz=f (x1, x2, . . . , xn), then the two second-order cross-partial derivativeszijandzj i are usually equal. That is,
∂
∂xj ∂f
∂xi
= ∂
∂xi ∂f
∂xj
This implies that the order of differentiation does not matter. The next theorem makes precise a more general result.
T H E O R E M 1 1 . 6 . 1 ( Y O U N G ’ S T H E O R E M )
Suppose that all themth-order partial derivatives of the functionf (x1, x2, . . . , xn) are continuous. If any two of them involve differentiating w.r.t. each of the vari- ables the same number of times, then they are necessarily equal.
The content of this result can be explained as follows: Letm=m1+ ã ã ã +mn, and suppose thatf (x1, x2, . . . , xn)is differentiatedm1 times w.r.t.x1,m2 times w.r.t.x2,. . ., andmn times w.r.t.xn. (Some of the integers m1, . . . , mn can be zero, of course.) Suppose that the continuity condition is satisfied for thesemth-order partial derivatives. Then we end up with the same result no matter what is the order of differentiation, because each of the final partial derivatives is equal to
∂mf
∂x1m1∂x2m2. . . ∂xnmn In particular, for the case whenm=2,
∂2f
∂xj∂xi = ∂2f
∂xi∂xj (i=1,2, . . . , n; j =1,2, . . . , n)
if both these partials are continuous. A proof of Young’s theorem is given in most advanced calculus books. Problem 11 shows that the mixed partial derivatives are not always equal.
Formal Definitions of Partial Derivatives
In Section 11.2, we gave a formal definition of partial derivatives for functions of two variables. This was done by modifying the definition of the (ordinary) derivative for a function of one variable. The same modification works for a function ofnvariables.
S E C T I O N 1 1 . 6 / P A R T I A L D E R I V A T I V E S W I T H M O R E V A R I A B L E S 403 Indeed, ifz=f (x1, . . . , xn), then withg(xi)=f (x1, . . . , xi−1, xi, xi+1, . . . , xn), we have∂z/∂xi =g(xi). (Here we think of all the variablesxj other thanxi as constants.) If we use the definition ofg(xi)(see (6.2.1), we obtain
∂z
∂xi =lim
h→0
f (x1, . . . , xi+h, . . . , xn)−f (x1, . . . , xi, . . . , xn)
h (4)
(The approximation in (2) holds because the fraction on the right in (4) is close to the limit ifh =0 is small enough.) If the limit in (4) does not exist, then we say that∂z/∂xi does not exist, or thatzis not differentiable w.r.t.xiat the point.
Virtually all the functions we consider have continuous partial derivatives everywhere in their domains. Ifz=f (x1, x2, . . . , xn)has continuous partial derivatives of first order in a domainD, we callf continuously differentiableinD. In this case,f is also called aC1 functiononD. If all partial derivatives up to orderkexist and are continuous,f is called aCkfunction.
P R O B L E M S F O R S E C T I O N 1 1 . 6
1. CalculateF1(1,1,1),F2(1,1,1), andF3(1,1,1)forF (x, y, z)=x2exz+y3exy.
⊂SM⊃2. Calculate all first-order partials of the following functions:
(a) f (x, y, z)=x2+y3+z4 (b) f (x, y, z)=5x2−3y3+3z4 (c) f (x, y, z)=xyz (d) f (x, y, z)=x4/yz
(e) f (x, y, z)=(x2+y3+z4)6 (f) f (x, y, z)=exyz
3. Letxandybe the populations of two cities anddthe distance between them. Suppose that the number of travellersT between the cities is given by
T =kxy/dn (kandnare positive constants) Find∂T /∂x,∂T /∂y, and∂T /∂d, and discuss their signs.
4. Letgbe defined for all(x, y, z)by
g(x, y, z)=2x2−4xy+10y2+z2−4x−28y−z+24 (a) Calculateg(2,1,1),g(3,−4,2), andg(1,1, a+h)−g(1,1, a).
(b) Find all partial derivatives of the first and second order.
5. Letπ(p, r, w)=14p2(1/r+1/w). Find the partial derivatives ofπw.r.t.p,r, andw.
6. Find all first- and second-order partials ofw(x, y, z)=3xyz+x2y−xz3. 7. Iff (x, y, z)=p(x)+q(y)+r(z), what aref1,f2, andf3?
8. Find the Hessian matrices of: (a) f (x, y, z)=ax2+by2+cz2 (b) g(x, y, z)=Axaybzc
9. Prove that ifw=
x−y+z x+y−z
h
, then x∂w
∂x +y∂w
∂y +z∂w
∂z =0
⊂SM⊃10. Calculate the first-order partial derivatives of the following function f (x, y, z)=xyz x >0, y >0, z >0 (You might find it easier first to take the natural logarithm of both sides.) HARDER PROBLEM
⊂SM⊃11. Define the functionf (x, y)=xy(x2−y2)/(x2+y2)when(x, y)=(0,0), andf (0,0)=0.
Find expressions forf1(0, y)andf2(x,0), then show thatf12(0,0)=1 andf21(0,0)= −1.
Check that Young’s theorem is not contradicted. (f12 andf21 are discontinuous at(0,0).)