Clairaut's Theorem says that if a function's mixed partial derivatives are continuous, then the order of differentiation does not matter. In Multivariable Calculus, that means fxy and fyx are equal when the smoothness conditions are met.
Clairaut's Theorem is the rule in Multivariable Calculus that lets you switch the order of mixed partial derivatives when those derivatives are continuous. If a function is smooth enough, then differentiating first with respect to x and then y gives the same result as differentiating first with respect to y and then x.
That sounds small, but it saves you from treating fxy and fyx as two separate answers every time. For a well-behaved function, both routes lead to the same second-order partial derivative. The theorem is often written as fxy = fyx, or in notation form as ∂²f/∂x∂y = ∂²f/∂y∂x.
The condition matters. Clairaut's Theorem does not say mixed partials are always equal for every function. It only guarantees equality when the mixed partial derivatives are continuous in the region you're working in. If a function has a corner, a jump, or some other rough behavior, the order of differentiation can fail to match.
A quick example is a polynomial like f(x,y) = x^2y + 3xy^2. This function is smooth everywhere, so the mixed partials should match. You can compute fx = 2xy + 3y^2, then fxy = 2x + 6y, or fy = x^2 + 6xy, then fyx = 2x + 6y. Same result, just two routes to get there.
This theorem is really about smoothness and consistency. In multivariable calculus, once you know a function has continuous second partial derivatives, you can move between derivative orders without worrying about changing the answer. That makes higher-order partial derivative work much cleaner, especially when you are checking formulas, simplifying expressions, or verifying a function behaves nicely.
Clairaut's Theorem shows up any time you work with second-order partial derivatives in Multivariable Calculus. It lets you simplify computations because you do not have to check both derivative orders separately when the function is smooth enough. Instead of treating fxy and fyx like two unrelated objects, you can use the theorem to confirm they match.
That matters in problem sets where you are asked to compute higher-order partial derivatives, compare two expressions, or verify that a function has continuous mixed partials. It also comes up when you study the gradient and the local shape of a surface, because second derivatives tell you how the surface bends in different directions.
The theorem also trains you to pay attention to conditions, not just formulas. A lot of multivariable calculus is like that: the rule works only when the function has the right smoothness. So if a problem asks whether mixed partials are equal, the real task is often checking continuity first, then using the theorem.
You will also see this idea in proofs and conceptual questions about why derivative order can be swapped for nice functions but not for rough ones. That makes Clairaut's Theorem a good checkpoint for whether you really understand partial derivatives as rates of change in more than one variable.
Keep studying Multivariable Calculus Unit 3
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view galleryPartial Derivative
Clairaut's Theorem starts with first-order partial derivatives. You take a partial derivative with respect to one variable while holding the others constant, and then you differentiate that result again. If the function is smooth, the theorem tells you the mixed second partials coming from those partial derivatives will agree.
Mixed Partial Derivative
This is the exact kind of derivative Clairaut's Theorem compares. A mixed partial derivative comes from differentiating with respect to two different variables, such as x then y. The theorem says the order of those two variables does not matter when the mixed partials are continuous.
Continuity
Continuity is the condition that makes Clairaut's Theorem work. If the mixed partial derivatives are continuous near the point you're checking, then the theorem guarantees equality. If continuity fails, you cannot assume the two mixed partials will match just because the notation looks similar.
Higher-Order Partial Derivatives
Clairaut's Theorem is part of the bigger topic of higher-order partial derivatives. Once you move past first derivatives, you start tracking second and sometimes third derivatives. The theorem gives you a shortcut by showing that some of those higher-order derivatives are interchangeable under the right smoothness conditions.
A quiz or free-response problem usually asks you to compute fxy and fyx, then decide whether they are equal. The move is simple: find the mixed partials in both orders, and check whether the function is smooth enough to justify Clairaut's Theorem. If the function is a polynomial or another smooth expression, you can usually state that the mixed partials are continuous and then use the theorem. If the function has a piecewise definition, a denominator that can hit zero, or another rough spot, you need to be more cautious and verify continuity before claiming the derivatives match.
A mixed partial derivative is the derivative you compute, like fxy or fyx. Clairaut's Theorem is the rule about when those two mixed partial derivatives are equal. So one is the object, and the other is the statement connecting the two objects.
Clairaut's Theorem says continuous mixed partial derivatives can be taken in either order without changing the result.
The notation fxy = fyx is shorthand for the theorem when the function is smooth enough.
You cannot use the theorem unless the mixed partial derivatives are continuous in the region you are checking.
A smooth polynomial usually satisfies the theorem, while a rough or piecewise function may not.
This theorem is a quick check on whether a multivariable function behaves nicely when you differentiate twice.
It is the rule that says the order of mixed partial differentiation does not matter when the mixed partial derivatives are continuous. So if a function is smooth enough, fxy and fyx are equal. That makes second-order partial derivative work much simpler.
You can use it when the mixed partial derivatives are continuous near the point you're studying. For many smooth functions, like polynomials, that condition is easy to satisfy. If the function is piecewise or has a singularity, you should not assume the theorem applies without checking.
A mixed partial derivative is the derivative itself, such as ∂²f/∂x∂y. Clairaut's Theorem is the result that says the mixed partials are equal when the continuity condition is met. The derivative is what you calculate, and the theorem is what lets you compare the two orders.
Compute the mixed partials in both orders and see whether they match. Then check whether the function is smooth enough for the theorem to apply. If the function is a polynomial or another nice formula, you can usually rely on the theorem after calculating.