Let be a function of . We have studied in great detail the derivative of with respect to , that is, , which measures the rate at which changes with respect to . Consider now . It makes sense to want to know how changes with respect to and/or . This section begins our investigation into these rates of change.
Consider the function , as graphed in Figure 13.3.2.(a). By fixing , we focus our attention to all points on the surface where the -value is 2, shown in both Figure 13.3.2.(a) and Figure 13.3.2.(b). These points form a curve in the plane : which defines as a function of just one variable. We can take the derivative of with respect to along this curve and find equations of tangent lines, etc.
The key notion to extract from this example is: by treating as constant (it does not vary) we can consider how changes with respect to . In a similar fashion, we can hold constant and consider how changes with respect to . This is the underlying principle of partial derivatives. We state the formal, limit-based definition first, then show how to compute these partial derivatives without directly taking limits.
Example 13.3.4 found a partial derivative using the formal, limit-based definition. Using limits is not necessary, though, as we can rely on our previous knowledge of derivatives to compute partial derivatives easily. When computing , we hold fixed — it does not vary. Therefore we can compute the derivative with respect to by treating as a constant or coefficient.
We have . Begin with . Keep fixed, treating it as a constant or coefficient, as appropriate:
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Note how the and terms go to zero. To compute , we hold fixed:
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Note how the and terms go to zero.
We have . Begin with . We need to apply the Chain Rule with the cosine term; is the coefficient of the -term inside the cosine function.
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To find , note that is the coefficient of the term inside of the cosine term; also note that since is fixed, is also fixed, and we treat it as a constant.
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We have . Beginning with , note how we need to apply the Product Rule.
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Note that when finding we do not have to apply the Product Rule; since does not contain , we treat it as fixed and hence becomes a coefficient of the term.
We have shown how to compute a partial derivative, but it may still not be clear what a partial derivative means. Given , measures the rate at which changes as only varies: is held constant.
Imagine standing in a rolling meadow, then beginning to walk due east. Depending on your location, you might walk up, sharply down, or perhaps not change elevation at all. This is similar to measuring : you are moving only east (in the “”-direction) and not north/south at all. Going back to your original location, imagine now walking due north (in the “”-direction). Perhaps walking due north does not change your elevation at all. This is analogous to : does not change with respect to . We can see that and do not have to be the same, or even similar, as it is easy to imagine circumstances where walking east means you walk downhill, though walking north makes you walk uphill.
It is also useful to note that . What does each of these numbers mean?
Consider , along with Figure 13.3.9.(a). If one “stands” on the surface at the point and moves parallel to the -axis (i.e., only the -value changes, not the -value), then the instantaneous rate of change is . Increasing the -value will decrease the -value; decreasing the -value will increase the -value.
Figure13.3.9.Illustrating the meaning of partial derivatives
Now consider , illustrated in Figure 13.3.9.(b). Moving along the curve drawn on the surface, i.e., parallel to the -axis and not changing the -values, increases the -value instantaneously at a rate of 1. Increasing the -value by 1 would increase the -value by approximately 1.
Since the magnitude of is greater than the magnitude of at , it is “steeper” in the -direction than in the -direction.
Another way to interpret partial derivatives is in terms of the tangent plane. Consider the graph of a function , such as the one in Figure 13.3.2. Setting , defines a point on the graph. Through the point , we have the lines , and , parallel to the and axes, respectively (where are parameters).
Now consider computing . The first two components of this derivative are found in a straightforward manner: they are and , respectively. To find the third component of the derivative, notice that in we vary the -component of while holding the -component constant. Using the Chain Rule and Definition 13.3.3, we find that the third component is . Altogether, we have
It seems reasonable that any vector that is tangent to these curves, which lie on our surface, should also be considered tangent to that surface. The vectors and are therefore tangent to at , and they are definitely not parallel. From Section 11.6 we know that any two non-parallel vectors at a point define a plane through that point. We also know that taking the cross product of these two vectors gives us a normal vector: the cross product gives us
Our function is , and we have , so the point on the surface is . The partial derivatives are and , so ,. Using Definition 13.3.10, our plane is given by
Notice the similarity between the tangent plane equation in Definition 13.3.10 and the single variable tangent line equation . As with functions of one variable, this suggests a connection between derivatives and linear approximation. We explore this connection in Section 13.4, where we’ll see that Definition 13.3.10 should be strengthed to require that the partial derivatives of be continuous.
Let . We have learned to find the partial derivatives and , which are each functions of and . Therefore we can take partial derivatives of them, each with respect to and . We define these “second partials” along with the notation, give examples, then discuss their meaning.
The notation of second partial derivatives gives some insight into the notation of the second derivative of a function of a single variable. If , then . The “” portion means “take the derivative of twice,” while “” means “with respect to both times.” When we only know of functions of a single variable, this latter phrase seems silly: there is only one variable to take the derivative with respect to. Now that we understand functions of multiple variables, we see the importance of specifying which variables we are referring to.
In each, we give and immediately and then spend time deriving the second partial derivatives.
Because the following partial derivatives get rather long, we omit the extra notation and just give the results. In several cases, multiple applications of the Product and Chain Rules will be necessary, followed by some basic combination of like terms.
Notice how in each of the three functions in Example 13.3.13, . Due to the complexity of the examples, this likely is not a coincidence. The following theorem states that it is not.
Again we refer back to a function of a single variable. The second derivative of is “the derivative of the derivative,” or “the rate of change of the rate of change.” The second derivative measures how much the derivative is changing. If , then the derivative is getting smaller (so the graph of is concave down); if , then the derivative is growing, making the graph of concave up.
Now consider . Similar statements can be made about and as could be made about above. When taking derivatives with respect to twice, we measure how much changes with respect to . If , it means that as increases, decreases, and the graph of will be concave down in the -direction. Using the analogy of standing in the rolling meadow used earlier in this section, measures whether one’s path is concave up/down when walking due east.
Similarly, measures the concavity in the -direction. If , then is increasing with respect to and the graph of will be concave up in the -direction. Appealing to the rolling meadow analogy again, measures whether one’s path is concave up/down when walking due north.
We now consider the mixed partials and . The mixed partial measures how much changes with respect to . Once again using the rolling meadow analogy, measures the slope if one walks due east. Looking east, begin walking north (side-stepping). Is the path towards the east getting steeper? If so, . Is the path towards the east not changing in steepness? If so, then . A similar thing can be said about : consider the steepness of paths heading north while side-stepping to the east.
The slope of the tangent line at in the direction of is : if one moves from that point parallel to the -axis, the instantaneous rate of change will be . The slope of the tangent line at this point in the direction of is : if one moves from this point parallel to the -axis, the instantaneous rate of change will be . These tangents lines are graphed in Figure 13.3.17.(a) and Figure 13.3.17.(b), respectively, where the tangent lines are drawn in a solid line.
Figure13.3.17.Understanding the second partial derivatives in Example 13.3.16
Now consider only Figure 13.3.17.(a). Three directed tangent lines are drawn (two are dashed), each in the direction of ; that is, each has a slope determined by . Note how as increases, the slope of these lines get closer to . Since the slopes are all negative, getting closer to 0 means the slopes are increasing. The slopes given by are increasing as increases, meaning must be positive.
Since , we also expect to increase as increases. Consider Figure 13.3.17.(b) where again three directed tangent lines are drawn, this time each in the direction of with slopes determined by . As increases, the slopes become less steep (closer to 0). Since these are negative slopes, this means the slopes are increasing.
Thus far we have a visual understanding of ,, and . We now interpret and . In Figure 13.3.17.(a), we see a curve drawn where is held constant at : only varies. This curve is clearly concave down, corresponding to the fact that . In part Figure 13.3.17.(b) of the figure, we see a similar curve where is constant and only varies. This curve is concave up, corresponding to the fact that .
Subsection13.3.5Partial Derivatives and Functions of Three Variables
The concepts underlying partial derivatives can be easily extend to more than two variables. We give some definitions and examples in the case of three variables and trust the reader can extend these definitions to more variables if needed.
We can continue taking partial derivatives of partial derivatives of partial derivatives of …; we do not have to stop with second partial derivatives. These higher order partial derivatives do not have a tidy graphical interpretation; nevertheless they are not hard to compute and worthy of some practice.
In the previous example we saw that ; this is not a coincidence. While we do not state this as a formal theorem, as long as each partial derivative is continuous, it does not matter the order in which the partial derivatives are taken. For instance, .
This can be useful at times. Had we known this, the second part of Example 13.3.20 would have been much simpler to compute. Instead of computing in the , then orders, we could have applied the , then then order (as ). It is easy to see that ; then and are clearly 0 as does not contain an or .
A brief review of this section: partial derivatives measure the instantaneous rate of change of a multivariable function with respect to one variable. With , the partial derivatives and measure the instantaneous rate of change of when moving parallel to the - and -axes, respectively. How do we measure the rate of change at a point when we do not move parallel to one of these axes? What if we move in the direction given by the vector ? Can we measure that rate of change? The answer is, of course, yes, we can. This is the topic of Section 13.6. First, we need to define what it means for a function of two variables to be differentiable.