First-Order Partial Derivatives

This section heavily relies on students to understand traces as the intersection of the surface plot of \(z=f(x,y)\) with a fundamental plane which allows us to simplify our multivariable problems into a single variable problem where we have lots of experience. This section also relates first partial derivatives to properties of multivariable functions on surface plots, contour plots, and tables. This section can be covered in a single class meeting but some instructors will pair this section with to talk about partial derivatives as a unified concept.

The last several activities offer a variety of applications and related approximation ideas for first partial derivatives. You may want to pick one of these and assign others as exercises.

The derivative plays a central role in single-variable calculus because it provides important information about a function in terms of each of the representations of the function. Thinking graphically, the derivative at a point tells us the slope of the line tangent to the graph at the given point. Numerically, the derivative at a point also provides the instantaneous rate of change of the function with respect to a change in the input variable. Algebraically, we can calculate the derivative as a function of the same input variable, which is useful as a tool to examine where the function is increasing or decreasing, as well as the concavity of the function's graph.

In this chapter, we are investigating functions of two or more variables and we can still ask how fast a function of several variables is changing, but we have to be careful about what we mean. Thinking graphically again, we can try to measure how steep the graph of the function is in a particular direction. As we saw in , the idea of measuring the behavior of a two variable function in a particular direction and generalizing to other directions is a much deeper idea when considering multiple input variables. Alternatively, we may want to know how quickly a function's output is changing in response to varying only one of the inputs. Over the next few sections, we will develop tools for addressing these questions. We begin with a Preview Activity that considers how we can apply our knowledge of single-variable calculus to measure the rate of change when we hold all but one input variable constant.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In Section, we studied the behavior of a function of two or more variables by considering the traces of the function. By holding one of our input variables constant, we restricted our multivariable function to have only one input variable, which means we can use all of our calculus tools from single-variable calculus on these traces.

Recall that in an earlier example, we considered the function \(f\) defined by \[\begin{aligned}\end{aligned}\]. This function measures horizontal distance (in feet) traveled by a projectile launched with an initial speed of \(x\) feet per second at an angle of \(y\) radians to the horizontal. The graph of this function is shown in Figure with traces plotted on the surface for \(x=150\) (in blue) and \(y=0.6\) (in red).

If we fix the angle \(y = 0.6\), we can view the trace \(f(x,0.6)\) as a function of \(x\) alone, as seen in Figure.

Since the trace is a function of only one variable, we can consider its derivative exactly as we did in single-variable calculus. With \(y=0.6\), we have \[\begin{aligned}\end{aligned}\] and therefore \[\begin{aligned}\end{aligned}\]. When \(x=150\), this is \[\begin{aligned}\end{aligned}\], which gives the slope of the tangent line shown in Figure.

Thinking of this derivative as an instantaneous rate of change implies that for a fixed launch angle of \(0.6\) radians, increasing the initial speed of the projectile by one foot per second leads to an expected increase in the horizontal distance traveled of approximately 8.74 feet. Since the sign of the derivative of the trace is positive there would be an increase to the range of the projectile if we increased the initial speed from \(150\, \text{ft/s}\) while holding the launch angle constant at \(0.6\) radians.

By holding \(y\) fixed and differentiating with respect to \(x\), we obtain the first-order partial derivative of \(f\) with respect to \(x\). Denoting this partial derivative as \(f_x\), we have seen that \[\begin{aligned}\end{aligned}\].

More generally, we have \[\begin{aligned}\end{aligned}\], provided this limit exists.

In the same way, we may obtain a trace by setting, say, \(x=150\) as shown in Figure.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Throughout single-variable calculus, you frequently used the fact that the derivative of a single-variable function can be interpreted geometrically as the slope of the line tangent to the graph at a given point. Similarly, we have seen that the partial derivatives measure the slope of a line tangent to a trace of a function of two variables as shown in Figure. Use the drop-down to change the trace and tangent line shown or show both traces and tangent lines.

Beyond representing the slope of the tangent line to the graph of a one-variable function or to a trace of a multivariable function, the values of derivatives and partial derivatives at a point give us the instantaneous rate of change of the function at that point. This perspective makes it possible for us to contemplate the units on partial derivatives, as the next activity asks you to investigate.

If you are tight on time, having students work on only the first part of during class and then completing the other parts for practice would be good. Being able to reason about the units of partial derivatives in applied contexts is an important skill for students to develop.

Notice that in this section, we did not need to use the classic calculus approach from scratch to define the partial derivative because we restricted our multivariable function along a trace to be a single-variable function. This was convenient, but we will soon see that we will need new tools to make similar arguments when studying the rate of change of a multivariable function in a direction that is not along a trace such as \(x=a\) or \(y=b\).

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.