12.3 partial derivatives

Recall that for a function f  of a single variable, we define the derivative function as
for any values of x for which the limit exists. At any particular value x = a, we interpret f ' (a) as the instantaneous rate of change of the function with respect to x at that point. For instance, if f (t ) represents the temperature of an object at time t, then f ' (t ) gives the rate of change of temperature with respect to time. In this section, we generalize the notion of derivative to functions of more than one variable.

Consider a flat metal plate in the shape of the region R 2. Suppose that the temperature at any point (x, y) R is given by f (x, y) . [Since the temperature only depends on the location (x, y) , recognize that this says that the temperature is independent of time.] A reasonable question might be to ask what the rate of change of f  is in the x - direction at a point (a, b) R. Think about it this way: if you move along the horizontal line segment from (a, b) to (a+h, b) , what is the average rate of change of the temperature with respect to the horizontal distance x (see Figure 12.19)? Notice that on this line segment, y is a constant ( y = b ). So, the average rate of change on this line segment is given by



Figure 12.19
Average temperature on a horizontal line segment.

To get the instantaneous rate of change of f  in the x - direction at the point (a, b), we take the limit as h 0 :
You should recognize this limit as a derivative. Since f  is a function of two variables and we have held the one variable fixed ( y = b ), we call this the partial derivative of f  with respect to x, denoted
This says that gives the instantaneous rate of change of f  with respect to x (i.e., in the x - direction) at the point (a, b) . Graphically, observe that in defining , we are looking only at points in the plane y = b . The intersection of z = f (x, y) and y = b is a curve, as shown in Figures 12.20a and 12.20b. Notice that the partial derivative then gives the slope of the tangent line to the curve at x = a , as indicated in Figure 12.20b.



Figure 12.20a
Intersection of the surface z = f (x, y) with the plane y = b.



Figure 12.20b
The curve z = f (x, b).



Figure 12.21
Average temperature on a vertical line segment.

Likewise, if we move along a vertical line segment from (a, b) to (a, b+h) (see Figure 12.21), the average rate of change of f  along this segment is given by

The instantaneous rate of change of f  in the y - direction at the point (a, b) is then given by
which you should again recognize as a derivative. In this case, however, we have held the value of x fixed ( x = a ) and refer to this as the partial derivative of f with respect to y, denoted
Graphically, observe that in defining , we are looking only at points in the plane x = a . The intersection of z = f (x, y) and x = a is a curve, as shown in Figures 12.22a and 12.22b (on the following page). In this case, notice that the partial derivative gives the slope of the tangent line to the curve at y = b , as shown in Figure 12.22b.

Graphical interpretations of second-order partial derivatives will be explored in the exercises.



Figure 12.22a
The intersection of the surface z = f (x, y) with the plane x = a.



Figure 12.22b
The curve z = f (a, y).

More generally, we define the partial derivative functions as follows.

3.1   
 
The partial derivative of f (x, y) with respect to x , written , is defined by
for any values of x and y for which the limit exists. The partial derivative of f (x, y) with respect to y , written , is defined by
for any values of x and y for which the limit exists.
 

Notation: Notice that since we are now dealing with functions of several variables, we can no longer use the same old prime notation for denoting partial derivatives. [Which partial derivative would f ' (x, y) denote?] We introduce several convenient types of notation here. For z = f (x, y), we write
The expression is a partial differential operator. It tells you to take the partial derivative (with respect to x ) of whatever expression follows it. Similarly, we have

Look carefully at how we defined these derivatives and you'll see that we can compute partial derivatives using all of our usual rules for computing ordinary derivatives. Notice that in the definition for the value of y is held constant, say at y = b . If we define g(x) = f (x, b) , then



That is, to compute the partial derivative , you simply take an ordinary derivative with respect to x , while treating y as a constant. Similarly, you can compute by taking an ordinary derivative with respect to y, while treating x as a constant.

3.1   
Computing Partial Derivatives
 
For f (x, y) = 3x2+x3y+4y2 , compute and f y(2, -1).
 
 
Compute by treating y as a constant (just like the coefficients 3 and 4). We have
Notice here that the partial derivative of 4y2 with respect to x is 0, since 4y2 is treated as if it were a constant when differentiating with respect to x. Next, we compute by treating x as a constant. We have
Substituting values in for x and y, we get
and

Again, since we are holding one of the variables fixed when we compute a partial derivative, we can use all of our familiar rules for computing derivatives. For instance, we have the product rules:
and
and the quotient rule:
with a corresponding quotient rule holding for

3.2   
Computing Partial Derivatives
 
For , compute and .
 
 
Recall that if , then , from the chain rule. Replacing the 4 with y and treating it as you would any other constant, we have
For the y - partial derivative, recall that if , then . Replacing the 4 with x and treating it as you would any other constant, we have

We interpret partial derivatives as rates of change, in the same way as we interpret ordinary derivatives of functions of a single variable.

3.3   
An Application of Partial Derivatives to Thermodynamics
 
For a real gas, van der Waals' equation states that
Here, P is the pressure of the gas, V is the volume of the gas, T is the temperature (in degrees Kelvin), n is the number of moles of gas, R is the universal gas constant and a and b are constants. Compute and interpret and .
 
 
We first solve for P to get
and compute
Notice that this gives the rate of change of pressure relative to a change in volume (with temperature held constant). Next, solving van der Waals' equation for T , we get
and compute
This gives the rate of change of temperature relative to a change in pressure (with volume held constant). You will discover an interesting fact about these partial derivatives in the exercises.
 

Notice that the partial derivatives found in the preceding examples are themselves functions of two variables. We have seen that second-and higher-order derivatives of functions of a single variable provide much significant information. Not surprisingly, higher-order partial derivatives are also very important in applications.

For functions of two variables, there are four different second-order partial derivatives. The partial derivative with respect to x of is , usually abbreviated as or f xx . Similarly, taking two successive partial derivatives with respect to y gives us . For mixed second-order partial derivatives, one derivative is taken with respect to each variable. If the first partial derivative is taken with respect to x , we have , abbreviated as , or (f x)y = f xy . If the first partial derivative is taken with respect to y , we have , abbreviated as , or .

3.4   
Computing Second-Order Partial Derivatives
 
Find all second-order partial derivatives of f (x, y) = x2y-y3+ln x.
 
 
We start by computing the first-order partial derivatives: and . We then have
and finally,

Notice in example 3.4 that . It turns out that this is true for most, but not all, of the functions that you will encounter. The proof of the following result can be found in most texts on advanced calculus.

3.1   
 
If f xy(x, y) and f yx(x, y) are continuous on an open set containing (a, b) , then f xy(a, b) = f yx(a, b) .
 

We can, of course, continue taking derivatives, computing third -, fourth-or even higher-order partial derivatives. Theorem 3.1 can be extended to show that as long as the partial derivatives are all continuous, the order of differentiation doesn't matter. With higher-order partial derivatives, notations such as become quite awkward and so, we usually use f xyx instead.

3.5   
Computing Higher-Order Partial Derivatives
 
For f (x, y) = cos(xy)-x3+y4 , compute f xyy and f xyyy .
 
 
We have
Differentiating f x with respect to y gives us
and
Finally, we have

Thus far, we have worked with partial derivatives of functions of two variables. The extensions to functions of three or more variables are completely analogous to what we have discussed here. In the exercises, you will be asked to write out definitions for partial derivatives of functions of three variables. In the following example, you can see that the calculations proceed just as you would expect.

3.6   
Partial Derivatives of Functions of Three Variables
 
For , defined for x, y, z 0, compute f x, f xy and f xyz .
 
 
To keep x , y and z as separate as possible, we first rewrite f  as
f (x, y, z) = x1/2y3/2z1/2+4x2y.
To compute the partial derivative with respect to x, we treat y and z as constants and obtain
Next, treating x and z as constants, we get
Finally, treating x and y as constants, we get
Notice that this derivative is defined for x, z > 0 and y 0 . Further, you can show that all first -, second-and third-order partial derivatives are continuous for x, z > 0 and y 0 , so the order in which we take the partial derivatives is irrelevant in this case.
 

3.7   
An Application of Partial Derivatives to a Sagging Beam
 
The sag in a beam of length L , width w and height h (see Figure 12.23) is given by for some constant c. Show that and . Use this result to determine which variable has the greatest proportional effect on the sag.



Figure 12.23
A horizontal beam.


 
 
We start by computing
We need to manipulate this expression to rewrite it in terms of S . Multiplying top and bottom by L , we get
The other calculations are similar and are left as exercises. To interpret the results, suppose that a small change L in length produces a small change S in the sag. We now have that Rearranging the terms, we have
That is, the proportional change in S is approximately four times the proportional change in L . Similarly, we have that in absolute value, the proportional change in S is approximately the proportional change in w and three times the proportional change in h . Proportionally then, a change in the length has the greatest effect on the amount of sag. In this sense, length is the most important of the three dimensions.
 

In many applications, no formula for the function is available and we can only estimate the value of the partial derivatives from a small collection of data points.

3.8   
Estimating Partial Derivatives from a Table of Data
 
A computer simulation of the flight of a baseball provided the data displayed in the following table for the range f (v, ) in feet of a ball hit with initial velocity ft/s and backspin rate of rpm. Each ball is struck at an angle of 30° above the horizontal.

v\ 0 1000 2000 3000 4000
150 294 312 333 350 367
160 314 334 354 373 391
170 335 356 375 395 414
180 355 376 397 417 436

Use the data to estimate and Interpret both quantities in baseball terms.
 
 
From the definition of partial derivative, we know that
so we can approximate the value of the partial derivative by computing the difference quotient for as small a value of h as possible. Since data points are provided for v = 150, we can compute the difference quotient for h = -10, to get
We can also use the data point for v = 170, to get
Since both estimates equal 2.1, we make the estimate The data point f (160, 2000) = 354 tells us that a ball struck with initial velocity 160 ft/s and backspin 2000 rpm will fly 354 feet. The partial derivative tells us that increasing the initial velocity by 1 ft/s will add approximately 2.1 feet to the distance.

We do similar computations to estimate noting that the closest data values to = 2000 are = 1000 and = 3000. We get

and
Reasonable estimates for are then 0.02, 0.019 or 0.0195 (the average of the two calculations). Using 0.02 as our approximation, we can interpret this to mean that an increase in backspin of 1 rpm will add approximately 0.02 ft to the distance. A simpler way to interpret this is to say that an increase of 100 rpm will add approximately 2 ft to the distance.
 

© 2002 McGraw-Hill Companies, Inc.