mixed partial derivatives
After differentiating with respect to one variable you can differentiate the result with respect to a different variable. A mixed partial derivative is the outcome of such a two-step process across two distinct variables — for instance, first nudge x and see how the slope-in-y changes. It probes how the function's slope in one direction varies as you move in another.
Formally, for f(x, y) the two second-order mixed partials are d^2 f / (dy dx), meaning differentiate first by x then by y, and d^2 f / (dx dy), meaning first by y then by x. (Notation conventions for the order vary between books, so read carefully.) For functions of more variables there is one mixed partial for each ordered pair of distinct variables.
The natural question — does the order of differentiation matter? — has a reassuring but conditional answer. In general the two orders can give different results when the function is badly behaved, but under a mild continuity hypothesis they always agree. That equality, Clairaut's theorem, is what lets us write second derivatives as a symmetric Hessian matrix without ambiguity.
f(x, y) = x^3 y^2. Then df/dx = 3x^2 y^2, and d/dy of that is 6x^2 y. Also df/dy = 2x^3 y, and d/dx of that is 6x^2 y. The two orders agree, as Clairaut promises for this smooth polynomial.
For smooth functions the two orders coincide.
Equality of mixed partials is not automatic. The standard counterexample f(x, y) = xy(x^2 - y^2)/(x^2 + y^2) (with f(0,0) = 0) has both mixed second partials at the origin, but they are unequal there — because they are not continuous at (0,0). Continuity is what Clairaut's theorem needs.