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If ∂²f/∂x∂y and ∂²f/∂y∂x are continuous at a point, then ∂²f/∂x∂y = ∂²f/∂y∂x. This 'equality of mixed partials' shows that (for most practical functions) the order of differentiation does not matter.
From your study of higher-order partial derivatives, you know that after taking a partial derivative of a multivariable function, the result is another function that you can differentiate again. The mixed partial derivatives ∂²f/∂x∂y and ∂²f/∂y∂x both measure how the function curves in both the x and y directions — but they arrive there by a different route. The first differentiates with respect to y first, then x; the second reverses the order. A natural question is whether the route matters.
Clairaut's theorem (also called Schwarz's theorem) says: if both mixed partials exist and are continuous at a point, they are equal there. For the smooth functions that appear in calculus courses — polynomials, exponentials, sines, cosines, and their combinations — continuity of the mixed partials is automatic, so the order of differentiation is irrelevant in practice. As a heuristic: if f is built from standard elementary functions and has no special piecewise behavior, assume the mixed partials commute.
To see why continuity matters, consider the classic counterexample: f(x, y) = xy(x² − y²)/(x² + y²) for (x, y) ≠ (0, 0) and f(0, 0) = 0. Careful computation shows ∂²f/∂x∂y|(0,0) = 1 while ∂²f/∂y∂x|(0,0) = −1. The mixed partials exist but are not equal because the continuity hypothesis fails at the origin. This example shows the theorem is not vacuous — the continuity condition is doing real work.
In practice, Clairaut's theorem means you can choose whichever order of differentiation is algebraically easier. When computing the Hessian matrix (the matrix of all second-order partial derivatives), the off-diagonal entries are mixed partials: H_ij = ∂²f/∂xᵢ∂xⱼ. Clairaut's theorem guarantees H is symmetric for smooth functions, which is a crucial structural fact — symmetric matrices have real eigenvalues, and the signs of those eigenvalues determine whether a critical point is a local min, local max, or saddle point. So this seemingly small commutativity result underpins the second derivative test in multiple dimensions.