Given a function f of n variables x1, x2,…, xn, the partial derivative ∂f/∂xi, where 1≤i≤n, may also be reckoned to be a function of x1, x2,…, xn. So the partial derivatives of ∂f/∂xi can be considered. Thuscan be formed, and these are denoted, respectively, byThese are the second-order partial derivatives. When j ≠ i, are different by definition, but the two are equal for most ‘straightforward’ functions f that are likely to be met. (It is not possible to describe here just what conditions are needed for equality.) Similarly, third-order partial derivatives such ascan be defined, as can fourth-order partial derivatives, and so on. Then the n-th-order partial derivatives, where n≥2, are called the higher-order partial derivatives.
When f is a function of two variables x and y, and the partial derivatives are denoted by fx and fy, then fxx, fxy, fyx, fyy are used to denoterespectively, noting particularly that fxy means (fx)y and fyx means (fy)x. This notation can be extended to third-order (and higher) partial derivatives and to functions of more variables. With the value of f at (x, y) denoted by f(x, y) and the partial derivatives denoted by f1 and f2, the second-order partial derivatives can be denoted by f11, f12, f21 and f22, and this notation can also be extended to third-order (and higher) partial derivatives and to functions of more variables.