APEX Calculus

Consider the function \(f(x,y) = x^2+2y^2\text{,}\) as graphed in Figure 13.3.2.(a). By fixing \(y=2\text{,}\) we focus our attention to all points on the surface where the \(y\)-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 \(y=2\text{:}\) \(z = f(x,2) = x^2+8\) which defines \(z\) as a function of just one variable. We can take the derivative of \(z\) with respect to \(x\) along this curve and find equations of tangent lines, etc.

The key notion to extract from this example is: by treating \(y\) as constant (it does not vary) we can consider how \(z\) changes with respect to \(x\text{.}\) In a similar fashion, we can hold \(x\) constant and consider how \(z\) changes with respect to \(y\text{.}\) 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 \(f_x(x,y)\text{,}\) we hold \(y\) fixed — it does not vary. Therefore we can compute the derivative with respect to \(x\) by treating \(y\) as a constant or coefficient.

Just as \(\frac{d}{dx}\big(5x^2\big) = 10x\text{,}\) we compute \(\frac{\partial}{\px}\big(x^2y\big) = 2xy\text{.}\) Here we are treating \(y\) as a coefficient.

Just as \(\frac{d}{dx}\big(5^3\big) = 0\text{,}\) we compute \(\frac{\partial}{\px}\big(y^3\big) = 0\text{.}\) Here we are treating \(y\) as a constant. More examples will help make this clear.

We have shown how to compute a partial derivative, but it may still not be clear what a partial derivative means. Given \(z=f(x,y)\text{,}\) \(f_x(x,y)\) measures the rate at which \(z\) changes as only \(x\) varies: \(y\) 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 \(z_x\text{:}\) you are moving only east (in the “\(x\)”-direction) and not north/south at all. Going back to your original location, imagine now walking due north (in the “\(y\)”-direction). Perhaps walking due north does not change your elevation at all. This is analogous to \(z_y=0\text{:}\) \(z\) does not change with respect to \(y\text{.}\) We can see that \(z_x\) and \(z_y\) 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.

The following example helps us visualize this more.

Another way to interpret partial derivatives is in terms of the tangent plane. Consider the graph of a function \(f(x,y)\text{,}\) such as the one in Figure 13.3.2. Setting \(x=a\text{,}\) \(y=b\) defines a point \((a,b,f(a,b))\) on the graph. Through the point \((a,b)\text{,}\) we have the lines \(x=a+s, y=b\text{,}\) and \(x=a, y=b+t\text{,}\) parallel to the \(x\) and \(y\) axes, respectively (where \(s,t\) are parameters).

Notice the similarity between the tangent plane equation in Definition 13.3.10 and the single variable tangent line equation \(y = f(c)+f'(c)(x-c)\text{.}\) 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 \(f\) be continuous.

Let \(z=f(x,y)\text{.}\) We have learned to find the partial derivatives \(f_x(x,y)\) and \(f_y(x,y)\text{,}\) which are each functions of \(x\) and \(y\text{.}\) Therefore we can take partial derivatives of them, each with respect to \(x\) and \(y\text{.}\) 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 \(y=f(x)\text{,}\) then \(\fp'(x) = \frac{d^2 y}{dx^2}\text{.}\) The “\(d^2y\)” portion means “take the derivative of \(y\) twice,” while “\(dx^2\)” means “with respect to \(x\) 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.

Notice how in each of the three functions in Example 13.3.13, \(f_{xy} = f_{yx}\text{.}\) Due to the complexity of the examples, this likely is not a coincidence. The following theorem states that it is not.

Finding \(f_{xy}\) and \(f_{yx}\) independently and comparing the results provides a convenient way of checking our work.

Now that we know how to find second partials, we investigate what they tell us.

Again we refer back to a function \(y=f(x)\) of a single variable. The second derivative of \(f\) 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 \(\fp'(x)\lt 0\text{,}\) then the derivative is getting smaller (so the graph of \(f\) is concave down); if \(\fp'(x) \gt 0\text{,}\) then the derivative is growing, making the graph of \(f\) concave up.

Now consider \(z=f(x,y)\text{.}\) Similar statements can be made about \(f_{xx}\) and \(f_{yy}\) as could be made about \(\fp'(x)\) above. When taking derivatives with respect to \(x\) twice, we measure how much \(f_x\) changes with respect to \(x\text{.}\) If \(f_{xx}(x,y)\lt 0\text{,}\) it means that as \(x\) increases, \(f_x\) decreases, and the graph of \(f\) will be concave down in the \(x\)-direction. Using the analogy of standing in the rolling meadow used earlier in this section, \(f_{xx}\) measures whether one’s path is concave up/down when walking due east.

Similarly, \(f_{yy}\) measures the concavity in the \(y\)-direction. If \(f_{yy}(x,y) \gt 0\text{,}\) then \(f_y\) is increasing with respect to \(y\) and the graph of \(f\) will be concave up in the \(y\)-direction. Appealing to the rolling meadow analogy again, \(f_{yy}\) measures whether one’s path is concave up/down when walking due north.

We now consider the mixed partials \(f_{xy}\) and \(f_{yx}\text{.}\) The mixed partial \(f_{xy}\) measures how much \(f_x\) changes with respect to \(y\text{.}\) Once again using the rolling meadow analogy, \(f_{x}\) 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, \(f_{xy} \gt 0\text{.}\) Is the path towards the east not changing in steepness? If so, then \(f_{xy}=0\text{.}\) A similar thing can be said about \(f_{yx}\text{:}\) consider the steepness of paths heading north while side-stepping to the east.

The following example examines these ideas with concrete numbers and graphs.

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.

By taking partial derivatives of partial derivatives, we can find second partial derivatives of \(f\) with respect to \(z\) then \(y\text{,}\) for instance, just as before.

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 \(f_{xxy} = f_{yxx}\text{;}\) 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, \(f_{xxy} = f_{xyx} = f_{yxx}\text{.}\)

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 \(f_{xyz}\) in the \(x\text{,}\) \(y\) then \(z\) orders, we could have applied the \(z\text{,}\) then \(x\) then \(y\) order (as \(f_{xyz} = f_{zxy}\)). It is easy to see that \(f_z = -\sin(z)\text{;}\) then \(f_{zx}\) and \(f_{zxy}\) are clearly 0 as \(f_z\) does not contain an \(x\) or \(y\text{.}\)

A brief review of this section: partial derivatives measure the instantaneous rate of change of a multivariable function with respect to one variable. With \(z=f(x,y)\text{,}\) the partial derivatives \(f_x\) and \(f_y\) measure the instantaneous rate of change of \(z\) when moving parallel to the \(x\)- and \(y\)-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 \(\la 2,1\ra\text{?}\) 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.