Partial Derivatives

Definition and computation

Higher partials and Clairaut

Interpretations and applications

The partials' information content

For f(x, y) = x^2 y + sin(xy), the partial f_x equals which expression?

Geometrically, f_x(a, b) measures which slope?

Clairaut's theorem states which equality, under continuity of the partials?

To compute f_x of f(x, y) = x^3 y^2, which operation does one perform?

A hiker walks a path x = x(t), y = y(t) on the hillside f(x, y). The rate df/dt is the total derivative, combining f_x and f_y with which other factors?

A function of two variables has two first partials and three distinct second partials under Clairaut. As a number, how many distinct second partials? Answer with a number.

The partial derivative symbol is a curly d. As a single character, what is it? Answer with the character.

The heat equation has the first time derivative on the left. As a number, what order? Answer with a number.

For the Cobb-Douglas production P = c L^alpha K^(1-alpha), the marginal product of labor dP/dL equals which expression?

The wave equation is second order in both variables. As a number, how many second derivatives appear? Answer with a number.