Clairaut's theorem
When you take a second derivative across two variables, you might worry that differentiating in x then y could differ from y then x. Clairaut's theorem is the comforting result that, as long as things are smooth enough, the order does not matter — the two mixed partials are simply equal. The book-keeping of differentiation is symmetric.
Precisely: if the mixed second partial derivatives d^2 f/(dy dx) and d^2 f/(dx dy) both exist in a neighborhood of a point and are continuous at that point, then they are equal there. (A common, slightly stronger hypothesis is that f is C^2 — twice continuously differentiable — which makes all mixed partials symmetric.) The result is often attributed jointly to Clairaut and Schwarz.
The continuity hypothesis is not decorative. There are functions whose mixed second partials both exist at a point yet take different values, precisely because those partials are discontinuous there. So the honest statement is conditional: equality of mixed partials holds under continuity, not unconditionally. When it does hold, the Hessian matrix is symmetric.
For f(x, y) = e^{xy}: df/dx = y e^{xy}, then d/dy gives e^{xy} + xy e^{xy}. The other order df/dy = x e^{xy}, then d/dx gives e^{xy} + xy e^{xy}. Equal, as guaranteed for this smooth (C-infinity) function.
Smooth functions: mixed partials match in either order.