Fxy

fxy denotes the mixed second partial derivative of a scalar function f(x,y) with respect to x and then y, written as ∂^2 f / ∂x ∂y. It is commonly abbreviated as f_{xy}. In two variables, this derivative is one entry in the Hessian matrix, which includes f_xx, f_xy, f_yx, and f_yy.

Computation: to obtain fxy, first differentiate f with respect to x to obtain f_x(x,y) = ∂f/∂x, then differentiate f_x with respect to y: f_xy(x,y) = ∂/∂y [f_x(x,y)]. If f has continuous second partial derivatives in a neighborhood of a point (f is C^2), then the mixed partials commute, so f_xy = f_yx at that point (Clairaut's theorem).

Examples: let f(x,y) = x^2 y. Then f_x = 2xy and f_xy = ∂/∂y(2xy) = 2x. Another example: f(x,y) = e^{x+y}. Then f_x = e^{x+y}, and f_xy = ∂/∂y e^{x+y} = e^{x+y}.

Applications: fxy appears in the analysis of surface curvature, in the study of partial differential equations, and in multivariable modeling to describe how changes in one variable influence the rate of change with respect to another variable. It also forms part of the Hessian matrix used to assess the local behavior of f near a point.

Caveats: if f_x is not differentiable with respect to y, fxy may not exist. If f_xy exists but f_yx ≠ f_xy, the function is not C^2 at that point. In well-behaved functions, both mixed partials are equal wherever defined.