If the function has a second-order continuous partial derivative and must be continuous, then f 12 = f2 1 is satisfied. When the second partial derivative is continuous, f 12 is equal to f2 1. For f(u, v), f is a binary function, and the second-order partial derivatives are f 1 1(uu), f 12(uv), f2 1(vu), f22(vv). Where f 12 and f2 1 are the same. Generally not, depending on the grading standard.
Partial derivative in x direction
There is a binary function z=f(x, y), the point (x0, y0) is a point in its domain d, y is fixed at y0, and let x have an increment △x at x0. Accordingly, the function z=f(x, y) has an increment (called partial increment to x) △z=f(x0+△x, y0)-f(x0, y0).
If the limit of the ratio of △z to △x exists when △x→0, then this limit value is called the partial derivative of the function z=f(x, y) to x at (x0, y0), which is denoted as f'x(x0, y0) or the function z=f(x, y) at (x0, y0).