Clairaut's Theorem — Use Clairaut's theorem for mixed partials - Multivariable Calculus

SECTION 1

Historical Context & Motivation

When you first learn about derivatives, you work with functions of a single variable — slope at a point on a curve. But real-world quantities often depend on more than one input. The temperature in a room depends on position and time; the pressure of a gas depends on temperature and volume. Mathematicians in the 1700s began asking: when you take partial derivatives with respect to different variables, does the order in which you differentiate change the answer?

This question led to one of the most elegant and practical results in multivariable calculus: Clairaut's Theorem. Named after the French mathematician Alexis Claude Clairaut, it guarantees that — under reasonable conditions — mixed partial derivatives are equal regardless of the order of differentiation. The theorem saves enormous effort in computation and forms the backbone of many advanced results in physics and engineering.

1713 • Alexis Clairaut Born

Clairaut was a child prodigy born in Paris. He read his first paper to the French Academy of Sciences at age 13 and became one of the youngest members ever elected.

1740 • Clairaut's Theorem Published

Clairaut formally stated that for functions with continuous second-order partial derivatives, the mixed partials are equal. This result appeared in his work on the shape of the Earth.

1823 • Schwarz's Rigorous Proof

German mathematician Hermann Schwarz provided a rigorous proof with precisely stated continuity conditions. The result is sometimes called the Schwarz theorem or the symmetry of second derivatives.

1900s • Widespread Use in Physics

Clairaut's Theorem became essential in thermodynamics (Maxwell relations), electromagnetism, and fluid dynamics — anywhere mixed partial derivatives appear in physical laws.

The central question Clairaut answered is deceptively simple: if you have a function f(x, y) and you first differentiate with respect to x, then with respect to y, do you get the same result as differentiating first with respect to y, then x? As we will see, the answer is yes — provided the mixed partials are continuous.

SECTION 2

Core Principles & Definitions

Before diving into the theorem itself, let's make sure the foundational vocabulary is solid. A partial derivative measures how a multivariable function changes when you vary one input while holding all the others constant. If f(x, y) is a function of two variables, the partial derivative with respect to x is written ∂f/∂x, and it treats y as a constant during differentiation.

Partial Derivative

The derivative of a multivariable function with respect to one variable, treating all other variables as constants. Written as ∂f/∂x or f_x.

Mixed Partial Derivative

A second-order derivative where you differentiate with respect to two different variables. For example, ∂²f/∂y∂x means: first differentiate with respect to x, then with respect to y.

Continuity Condition

Clairaut's Theorem requires that the second-order mixed partials be continuous on the region in question. Most functions you encounter in practice satisfy this condition.

Equality of Mixed Partials

When the continuity condition holds, ∂²f/∂x∂y = ∂²f/∂y∂x. The order of differentiation does not matter — this is the core statement of Clairaut's Theorem.

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Notation Alert

Watch the order carefully! In the notation ∂²f/∂y∂x, you differentiate with respect to x first (rightmost), then y (leftmost). In subscript notation, f_xy means differentiate with respect to x first, then y. The subscript order reads left to right, while the fraction notation reads right to left.

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KEY TAKEAWAY

Think of mixed partial derivatives like driving directions. Imagine you need to go 3 blocks east and 2 blocks north. Whether you go east first then north, or north first then east, you end up at the same destination. Clairaut's Theorem says the same thing about differentiation: for well-behaved functions, the final answer is the same no matter which variable you differentiate first.

SECTION 3

Visual Explanation

The diagram below illustrates the two paths to a mixed partial derivative. Starting from the original function f(x, y), you can follow either branch — differentiating first with respect to x or first with respect to y — and Clairaut's Theorem guarantees that both paths lead to the same result.

Two differentiation paths from f(x, y) to a mixed second partial. Path A (cyan) differentiates with respect to x first, then y. Path B (pink) differentiates with respect to y first, then x. The green dashed line shows that both results are equal.

Notice that the two boxes at the bottom contain different-looking expressions — ∂²f/∂y∂x on the left and ∂²f/∂x∂y on the right — yet Clairaut's Theorem tells us they evaluate to the same function. This symmetry is not obvious from the notation alone, which is why the theorem is so powerful: it lets you choose whichever differentiation order is computationally easier, knowing the result will be the same.

SECTION 4

Mathematical Framework

Let's formalize what Clairaut's Theorem says and break down the notation. Understanding the precise statement will help you apply it with confidence.

CLAIRAUT'S THEOREM

If f(x, y) has continuous second partial derivatives, then f_xy = f_yx equivalently: ∂²f/∂y∂x = ∂²f/∂x∂y

f_xy means: differentiate f with respect to x first, then with respect to y. f_yx means: differentiate f with respect to y first, then with respect to x.

SUBSCRIPT NOTATION

f_xy = ∂/∂y ( ∂f/∂x ) and f_yx = ∂/∂x ( ∂f/∂y )

Read subscript notation left to right: f_xy differentiates with respect to x, then y. In fraction notation ∂²f/∂y∂x, read right to left: differentiate with respect to x first (innermost), then y.

EXAMPLE COMPUTATION

f(x, y) = x³y² + 2xy → f_x = 3x²y² + 2y → f_xy = 6x²y + 2

First we treat y as a constant and differentiate with respect to x. Then we treat x as a constant and differentiate the result with respect to y.

VERIFICATION (OTHER ORDER)

f(x, y) = x³y² + 2xy → f_y = 2x³y + 2x → f_yx = 6x²y + 2

Now we differentiate with respect to y first, then x. The result is 6x²y + 2 — identical to f_xy, confirming Clairaut's Theorem.

The key condition is that the second-order mixed partial derivatives must be continuous in a neighborhood of the point where you are evaluating them. In practice, this means the function should be "smooth" — no sharp corners, jumps, or undefined patches. Polynomial, exponential, trigonometric, and logarithmic functions (where defined) all meet this requirement.

SECTION 5

Detailed Breakdown & Surface Visualization

To build deeper intuition, consider what mixed partial derivatives measure geometrically. The first partial ∂f/∂x tells you the slope of the surface z = f(x, y) in the x-direction. When you then take ∂/∂y of that slope, you are measuring how fast that x-slope changes as y varies. Clairaut's Theorem says this cross-rate of change is the same whether you measure the x-rate of the y-slope or the y-rate of the x-slope.

A 3D surface z = x²y with tangent slopes at a point. The pink arrow shows the x-slope (∂f/∂x), and the amber arrow shows the y-slope (∂f/∂y). The mixed partial measures how each slope changes as the other variable moves.

For the function f(x, y) = x²y shown in the diagram, the first partials are f_x = 2xy and f_y = x². The mixed partials are f_xy = 2x and f_yx = 2x — indeed equal, just as the theorem predicts.

Function f(x, y)	f_xy	f_yx	Equal?
x³y² + 2xy	6x²y + 2	6x²y + 2	✓ Yes
sin(xy)	cos(xy) − xy sin(xy)	cos(xy) − xy sin(xy)	✓ Yes
e^x+y	e^x+y	e^x+y	✓ Yes
x²y³ − 4xy	6xy² − 4	6xy² − 4	✓ Yes

Verification of Clairaut's Theorem for common functions

SECTION 6

Worked Example

Let's work through a complete example using Clairaut's Theorem to verify that the mixed partials of a function are equal, and then use the theorem to simplify a computation.

Verify Clairaut's Theorem for f(x, y) = x²sin(y) + 3xy²

1
Step 1 — Find $f_x$ (differentiate with respect to x, treating y as constant)

Differentiate each term with respect to x: the derivative of x²sin(y) is 2x sin(y) (since sin(y) is constant with respect to x), and the derivative of 3xy² is 3y².

f_x = 2x sin(y) + 3y²
2
Step 2 — Find f_xy (differentiate $f_x$ with respect to y)

Now differentiate f_x with respect to y: the derivative of 2x sin(y) with respect to y is 2x cos(y), and the derivative of 3y² with respect to y is 6y.

f_xy = 2x cos(y) + 6y
3
Step 3 — Find $f_y$ (differentiate with respect to y, treating x as constant)

Go back to the original function and differentiate with respect to y: the derivative of x²sin(y) is x²cos(y), and the derivative of 3xy² is 6xy.

f_y = x²cos(y) + 6xy
4
Step 4 — Find f_yx (differentiate $f_y$ with respect to x)

Differentiate f_y with respect to x: the derivative of x²cos(y) is 2x cos(y), and the derivative of 6xy is 6y.

f_yx = 2x cos(y) + 6y
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Step 5 — Compare and Conclude

We found f_xy = 2x cos(y) + 6y and f_yx = 2x cos(y) + 6y. Since both mixed partials are equal, Clairaut's Theorem is confirmed. Both expressions are continuous everywhere, so the continuity condition is satisfied.

f_xy = f_yx = 2x cos(y) + 6y ✓

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Pro Tip

In many problems, one order of differentiation is easier than the other. If you need to find f_xy and differentiating with respect to y first gives a simpler expression, Clairaut's Theorem lets you compute f_yx instead and get the same answer. This can save you a lot of algebra!

SECTION 7

When Clairaut's Theorem Applies (and When It Doesn't)

Clairaut's Theorem is remarkably reliable, but it's important to understand its boundaries. The theorem requires continuity of the mixed partial derivatives. For the vast majority of functions you encounter in a calculus course — polynomials, rational functions (where defined), trigonometric, exponential, and logarithmic functions — this condition is automatically met.

Situation	Does the theorem apply?	Why?
Polynomial function	✓ Always	All partial derivatives of polynomials are continuous everywhere.
Trigonometric function	✓ Always	Sine, cosine, and their derivatives are continuous everywhere.
e raised to a polynomial	✓ Always	Exponential functions and their derivatives are continuous everywhere.
Piecewise-defined function	⚠ Maybe	Must check continuity of mixed partials at the boundary between pieces.
Function with a discontinuity	✗ No (at that point)	If the mixed partials are discontinuous, the theorem fails and f_xy ≠ f_yx is possible.

When does Clairaut's Theorem apply?

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PRACTICAL RULE

If you can write the function using a formula involving standard operations (addition, multiplication, powers, trig, exponentials, logs), and there are no division-by-zero issues or piecewise definitions near your point of interest, then Clairaut's Theorem will apply. In a typical high school or undergraduate calculus course, you can safely assume the theorem holds for every function you are asked to work with, unless the problem specifically tells you otherwise.

SECTION 8

Extension to Higher Dimensions & Advanced Theory

Clairaut's Theorem extends naturally beyond two variables. If you have a function f(x, y, z), the theorem tells you that any pair of mixed partials can be swapped. For instance, f_xz = f_zx, f_yz = f_zy, and f_xy = f_yx, provided the relevant mixed partials are continuous. Moreover, for third-order and higher derivatives, you can rearrange the differentiation order freely.

Concept	Clairaut's Theorem (2 variables)	Extended Version (3+ variables)
Statement	f_xy = f_yx	f_xy = f_yx, f_xz = f_zx, f_yz = f_zy, etc.
Number of mixed pairs	1 pair	3 pairs for 3 variables, 6 for 4, etc.
Higher-order extension	f_xxy = f_xyx = f_yxx	Any permutation of differentiation gives the same result
Applications	2D optimization, exact equations	Thermodynamics (Maxwell relations), electromagnetism, PDEs

Clairaut's Theorem in 2 variables vs. higher dimensions

In physics, Clairaut's Theorem plays a central role in thermodynamics through the Maxwell relations, which connect quantities like temperature, entropy, pressure, and volume. It also appears in differential equations, where testing whether a differential equation is exact requires checking the equality of certain mixed partials — a direct application of Clairaut's Theorem. These connections will become important if you go on to study AP Physics, engineering, or advanced mathematics.

SECTION 9

Practice Problems

Test your understanding with these five problems, arranged from conceptual to challenging. Try each problem on your own before checking the answer.

Question 1: CONCEPTUAL

In your own words, what does Clairaut's Theorem guarantee about a smooth function f(x, y)? What condition must the function satisfy for the theorem to apply?

Show answer

Clairaut's Theorem guarantees that for a function f(x, y) whose second-order mixed partial derivatives are continuous, the order of differentiation does not matter: f_xy = f_yx. The key condition is that both ∂²f/∂x∂y and ∂²f/∂y∂x must be continuous on the region of interest. This means the function must be 'smooth' — no sudden jumps, corners, or undefined behavior in its second derivatives.

Question 2: BASIC CALCULATION

Let f(x, y) = 4x³y − 5xy⁴. Compute f_xy and f_yx and verify they are equal.

Show answer

First find f_x = 12x²y − 5y⁴. Then f_xy = 12x² − 20y³. Now find f_y = 4x³ − 20xy³. Then f_yx = 12x² − 20y³. Both mixed partials equal 12x² − 20y³, confirming Clairaut's Theorem. Since f is a polynomial, the continuity condition is automatically satisfied.

Question 3: INTERMEDIATE

Let f(x, y) = e^2x cos(y). Find ∂²f/∂y∂x. Then, using Clairaut's Theorem, state the value of ∂²f/∂x∂y without computing it from scratch.

Show answer

First compute ∂f/∂x = 2e^2x cos(y). Then ∂²f/∂y∂x = ∂/∂y [2e^2x cos(y)] = −2e^2x sin(y). By Clairaut's Theorem, since e^2x cos(y) has continuous partial derivatives everywhere, we know ∂²f/∂x∂y = −2e^2x sin(y) without needing to redo the calculation. This saves you from differentiating ∂f/∂y = −e^2x sin(y) with respect to x separately.

Question 4: APPLIED

The temperature T at a point on a metal plate is given by T(x, y) = 100 − x² − 2y², where x and y are in centimeters and T is in degrees Celsius. Compute both mixed partials T_xy and T_yx. Interpret what the result means physically.

Show answer

T_x = −2x and T_y = −4y. Then T_xy = ∂/∂y(−2x) = 0, and T_yx = ∂/∂x(−4y) = 0. Both are equal to zero, confirming Clairaut's Theorem. Physically, this means the rate at which the x-temperature-gradient changes as you move in the y-direction is zero — the x and y temperature variations are independent of each other. This makes sense because each variable appears in a separate term of the temperature function.

Question 5: CRITICAL THINKING

Suppose you know that for a certain function g(x, y), the partial derivative g_x = 3x²y + e^y and g_y = x³ + xe^y + 2y. Use Clairaut's Theorem to determine whether these two expressions could genuinely come from the same function g(x, y). Explain your reasoning.

Show answer

If g_x = 3x²y + e^y and g_y = x³ + xe^y + 2y, then by Clairaut's Theorem we need g_xy = g_yx. Compute g_xy = ∂/∂y(3x²y + e^y) = 3x² + e^y. Compute g_yx = ∂/∂x(x³ + xe^y + 2y) = 3x² + e^y. Since g_xy = g_yx = 3x² + e^y, the Clairaut consistency check passes, and yes, these could come from the same function. (In fact, g(x, y) = x³y + xe^y + y² + C.) This technique is used to test whether a differential equation is exact.

SUMMARY

Lesson Summary

Summary

Clairaut's Theorem states that for a function f(x, y) with continuous second-order mixed partial derivatives, the order of differentiation does not matter: f_xy = f_yx. This means ∂²f/∂y∂x = ∂²f/∂x∂y. The theorem applies to all polynomials, trigonometric, exponential, and logarithmic functions (where they are defined), which covers virtually every function encountered in standard calculus courses.

Named after the French mathematician Alexis Claude Clairaut (1713–1765), the theorem allows you to choose the computationally easier differentiation order when finding mixed partials, and it serves as a consistency check for verifying partial derivative computations. It extends to functions of three or more variables and forms the foundation for advanced results in thermodynamics (Maxwell relations), differential equations (exactness tests), and mathematical analysis.