Second-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 and uses several ideas related to second derivatives from single variable calculus (concavity and increasing/decreasing slopes). This section can be covered in a single class meeting and some instructors pair this with the material on first partial derivatives from . The geometric meaning of the different second derivatives is used in the development of the optimization tools for functions of two variables in . In particular, goes through tasks related to concavity and twisting in a surface that is used to identify saddle points.

This section does not appeal to the classic calculus approach because we continue to use our tools from single variable calculus along traces of different kinds.

will be an important activity for helping students to understand the form of the discriminant in .

Recall that for a single-variable function \(f\), the second derivative of \(f\) is defined to be the derivative of the first derivative. That is, \(f''(x) = \frac{d}{dx}\Bigl[ f'(x) \Bigr] = \frac{d^2f}{dx^2}\), which can be stated in terms of the limit definition of the derivative by writing \[\begin{aligned}\end{aligned}\]

In the following preview activity, we will explore how to calculate the four different second-order partial derivatives of a function of two variables as well as what these various derivatives tell us about the function's behavior.

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

A function \(f\) of two variables \(x\) and \(y\) has two first-order partial derivatives, \(f_x=\frac{\partial f}{\partial x}\) and \(f_y=\frac{\partial f}{\partial y}\). As we saw in Preview Activity, each of these first-order partial derivatives has two partial derivatives, giving a total of four second-order partial derivatives: \[\begin{aligned}f_{xx} \amp= (f_x)_x = \frac{\partial}{\partial x} \Bigl[ \frac{\partial f}{\partial x} \Bigr] = \frac{\partial^2 f}{\partial x^2} \\ f_{yy} \amp= (f_y)_y=\frac{\partial}{\partial y} \Bigl[ \frac{\partial f}{\partial y} \Bigr] = \frac{\partial^2 f}{\partial y^2} \\ f_{xy} \amp= (f_x)_y=\frac{\partial}{\partial y} \Bigl[ \frac{\partial f}{\partial x} \Bigr] = \frac{\partial^2 f}{\partial y \partial x} \\ f_{yx}\amp=(f_y)_x=\frac{\partial}{\partial x} \Bigl[ \frac{\partial f}{\partial y} \Bigr] = \frac{\partial^2 f}{\partial x \partial y}\end{aligned}\]

We call \(f_{xx}\) and \(f_{yy}\) unmixed second-order partial derivatives. We refer to \(f_{xy}\) and \(f_{yx}\) as the mixed second-order partial derivatives.

You may need some practice to get accustomed to the notation for partial derivatives. By writing \[\begin{aligned}\end{aligned}\], we mean that we first differentiate with respect to \(x\) and then differentiate with respect to \(y\). This can be expressed in the alternate notation \(f_{xy} = (f_x)_y\). However, to find the second partial derivative \[\begin{aligned}\end{aligned}\], we first differentiate with respect to \(y\) and then differentiate with respect to \(x\). This means that \[\begin{aligned}\end{aligned}\].

The Leibniz notation and subscript notation for a mixed partial differ in terms of the order in which \(x\) and \(y\) appear in each notation. However, in both notations, we use the idea of order of operations working from the inside out. With the Leibniz notation, that means our partial derivative operators are written before the function we are taking partial derivatives of, whereas with the subscript notation writes order of derivatives after the function.

In Preview Activity and Activity, you may have noticed that the mixed second-order partial derivatives are equal. This observation holds generally and is known as Clairaut's Theorem.

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

Recall from single-variable calculus, that the first derivative describes whether a function is increasing or decreasing, but the second derivative measures how the function is increasing or decreasing.

In , the left curve has the slopes of the tangent lines increasing as you move along the graph from left to right, which means that the function is increasing at an increasing rate. In other words, the derivative of the slope of the tangent lines is positive, making \(f''(x) \gt 0\). We say that this curve is concave up. The middle curve in has tangent lines with the same slope, which means that the function is increasing at a constant rate. Thus the derivative of the slope of the tangent lines will be zero. The right curve in has the slope of the tangent lines decreasing as you move along the graph from left to right, which means the function is increasing at a decreasing rate. In other words, the derivative of the slope of the tangent line is negative, so \(f''(x) \lt 0\). We say that this curve is concave down.

A similar argument can be applied to the graphs of decreasing functions, as shown in . In particular, we can say that the left plot will have negative slopes of tangent lines but those slopes are increasing from left to right, so \(f''(x) \gt 0\) and the curve is concave up. The middle plot in has a constant slope for tangent lines along the curve, thus the second derivative is zero. The right plot in has negative slopes of tangent lines, corresponding to a decreasing graph, and the slopes of those tangent lines are decreasing. Thus \(f''(x) \lt 0\) and we say that the graph is concave down.

Remember that \(-10 \lt -5\) and \(-5 \lt 2\), so an increase in negative numbers means the numbers are getting closer to zero, while a decrease in negative numbers means that values are getting more negative or farther from zero.

Since the unmixed second-order partial derivative \(f_{xx}\) requires us to hold \(y\) constant and differentiate twice with respect to \(x\), we can view \(f_{xx}\) as the second derivative of a trace of \(f\) where \(y\) is fixed. As such, \(f_{xx}\) will measure the concavity of this trace.

In the following activity, we further explore what second-order partial derivatives tell us about the geometric behavior of a surface.

Just as with the first-order partial derivatives, we can approximate second-order partial derivatives in the situation where we have information about the function only at certain inputs, like with tables or contour plots.

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