Difficulty: Intermediate | Prerequisites: First-order partial derivatives (see companion notes).
Once you can take a first partial derivative, you can take the derivative of that result, giving you a second partial derivative. A function of two variables has four second-order partial derivatives. The most important result in this section is Clairaut's Theorem, which tells you that the two mixed partials are equal under standard conditions, saving you half the work.
Second partial derivatives are what you get when you differentiate a partial derivative again with respect to the same or a different variable. A two-variable function has four of them. Clairaut's Theorem says the two mixed second partials (f_xy and f_yx) are equal whenever they are continuous, which is nearly always the case with functions you meet in this course.
Second partial derivative
The partial derivative of a first partial derivative. For f(x, y), the second partial derivative f_xx means: first differentiate f with respect to x, then differentiate the result with respect to x again.
Think of it as: the same idea as the second derivative from Calculus I (used for concavity and the second derivative test), extended to functions with more than one input.
Mixed partial derivative (f_xy, f_yx)
A second partial derivative where you differentiate with respect to two different variables in sequence. f_xy means: first differentiate with respect to x, then with respect to y. f_yx means: first with respect to y, then with respect to x.
In simple terms, this means: you are asking how the rate of change in one direction itself changes as you move in a perpendicular direction.
Clairaut's Theorem (equality of mixed partials)
If f is defined on an area around a point (a, b) and both f_xy and f_yx are continuous on that area, then f_xy = f_yx at every point in that area.
Think of it as: the order in which you take the two partial derivatives does not matter, provided the function is reasonably well behaved (and it will be, for every function you meet in this course).
For a function f(x, y) of two variables, there are four second-order partial derivatives:
f_xx: differentiate f with respect to x, then differentiate the result with respect to x again.
f_xy: differentiate f with respect to x, then differentiate the result with respect to y.
f_yx: differentiate f with respect to y, then differentiate the result with respect to x.
f_yy: differentiate f with respect to y, then differentiate the result with respect to y again.
The process is mechanical: find the first partials, then treat each first partial as a new function and take its partial derivatives.
f(x, y) = x^3 + x^2 y^3 - 2y^2
First partials:
f_x = 3x^2 + 2x y^3
f_y = 3x^2 y^2 - 4y
Second partials:
f_xx = 6x + 2y^3 (differentiate f_x with respect to x)
f_xy = 6x y^2 (differentiate f_x with respect to y)
f_yx = 6x y^2 (differentiate f_y with respect to x)
f_yy = 6x^2 y - 4 (differentiate f_y with respect to y)
Notice that f_xy = f_yx = 6x y^2, exactly as Clairaut's Theorem predicts.
Clairaut's Theorem is not just a theoretical nicety. It gives you a shortcut: if you need a mixed partial like f_xyz for a complicated function, you can rearrange the order of differentiation to whichever sequence is easiest.
Consider f(x, y, z) = x y^2 z^3 + arcsec(x sqrt(z)). Finding f_xyz directly means differentiating the arcsec term three times with increasingly messy expressions. Clairaut's Theorem lets you compute f_yzx instead:
f_y = 2x y z^3 (the arcsec term has no y, so it drops out)
f_yz = 6x y z^2
f_yzx = 6y z^2
The arcsec term vanished at the first step, making the problem far simpler.
For z = f(x, y), these all mean f_xx:
f_xx
(partial / partial x)(partial f / partial x)
partial^2 f / partial x^2
partial^2 z / partial x^2
And these all mean f_xy (differentiate with respect to x first, then y):
f_xy
(partial / partial y)(partial f / partial x)
partial^2 f / (partial y partial x)
partial^2 z / (partial y partial x)
Note the Leibniz notation reads right to left: partial^2 f / (partial y partial x) means "first x, then y."
f_{xx} = \frac{\partial}{\partial x}\!\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^2 f}{\partial x^2}f_{xy} = \frac{\partial}{\partial y}\!\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^2 f}{\partial y\,\partial x}Clairaut's Theorem:
f_{xy} = f_{yx}whenever both mixed partials are continuous on the region in question.
Students often assume f_xy and f_yx are always equal without checking conditions. They are equal when both are continuous, which covers nearly every function in this course, but the theorem does have a hypothesis. If an exam question asks you to state the theorem, include the continuity condition.
The Leibniz notation for mixed partials reads right to left, which trips students up. In partial^2 f / (partial y partial x), you differentiate with respect to x first, then y. The subscript notation f_xy reads left to right: first x, then y. Both describe the same operation, but the reading direction is reversed.
Students sometimes believe that second partial derivatives are only useful for finding concavity. In multivariable calculus, they also appear in the second derivative test for local extrema (the D-test using f_xx, f_yy and f_xy), in Taylor expansions, and in partial differential equations.
The second derivative test for multivariable functions (the D-test) uses f_xx, f_yy and f_xy. You cannot apply it without being comfortable computing all four second partials.
Exam questions commonly ask you to find all second partial derivatives of a given function and verify that f_xy = f_yx. This tests both computation skill and knowledge of Clairaut's Theorem.
Knowing you can rearrange the differentiation order is a time-saver on exam problems involving complicated expressions. Look for the order that eliminates the most difficult terms earliest.
Be ready to state Clairaut's Theorem precisely, including its continuity hypothesis.
True or false: A function f(x, y) has exactly two second-order partial derivatives.
False. It has four: f_xx, f_xy, f_yx, and f_yy.
Fill in the blank: Clairaut's Theorem states that f_xy = f_yx whenever both mixed partials are ______ on the region.
Continuous.
True or false: In the notation partial^2 f / (partial y partial x), you differentiate with respect to y first.
False. You differentiate with respect to x first (read right to left), then y.
Fill in the blank: For f(x, y) = x^3 + x^2 y^3 - 2y^2, f_yy = ______.
6x^2 y - 4.
Q: Find all four second partial derivatives of f(x, y) = x^3 + x^2 y^3 - 2y^2.
A: f_x = 3x^2 + 2x y^3, f_y = 3x^2 y^2 - 4y. Then f_xx = 6x + 2y^3, f_xy = 6x y^2, f_yx = 6x y^2, f_yy = 6x^2 y - 4.
Q: For f(x, y, z) = x y^2 z^3 + arcsec(x sqrt(z)), find f_xyz. State any theorem you use.
A: By Clairaut's Theorem (both mixed partials continuous where f is defined), f_xyz = f_yzx. Then f_y = 2x y z^3, f_yz = 6x y z^2, f_yzx = 6y z^2. The arcsec term drops out at the first step because it does not contain y.
Q: Explain why Clairaut's Theorem is useful for computation, not just a theoretical result.
A: It lets you choose the easiest differentiation order. If one order forces you through a complicated expression (e.g. the derivative of arcsec), you can swap the order to one that simplifies or eliminates that term.
Second partial derivatives are essential for the multivariable second derivative test (the D-test), which you will use to classify critical points as local maxima, local minima, or saddle points. They also appear in Taylor series for multivariable functions and in partial differential equations such as the heat equation and wave equation.
Clairaut's Theorem generalises to higher-order mixed partials: you can rearrange any sequence of continuous partial derivatives into any other order. This becomes especially handy in topics like Laplacians and differential geometry.
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