Clairaut's theorem

Clairaut's theorem states that for a well-behaved function with continuous second derivatives, the mixed partial derivatives are equal regardless of the order of differentiation. In other words, if you differentiate a function with respect to two variables, say x and y, then differentiate twice, switching the order (first with respect to y, then x, or vice versa), the results are the same. This symmetry helps mathematicians simplify complex problems involving multiple variables and ensures consistency when analyzing how functions change across different directions.