## Quick Reference

Given a function *f* of *n* variables *x*_{1}, *x*_{2},…, *x*_{n}, the partial derivative ∂*f*/∂*x*_{i}, where 1≤*i*≤*n*, may also be reckoned to be a function of *x*_{1}, *x*_{2},…, *x*_{n}. So the partial derivatives of ∂*f*/∂*x*_{i} 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 *f*_{x} and *f*_{y}, then *f*_{xx}, *f*_{xy}, *f*_{yx}, *f*_{yy} are used to denoterespectively, noting particularly that *f*_{xy} means (*f*_{x})_{y} and *f*_{yx} means (*f*_{y})_{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 *f*_{1} and *f*_{2}, the second-order partial derivatives can be denoted by *f*_{11}, *f*_{12}, *f*_{21} and *f*_{22}, and this notation can also be extended to third-order (and higher) partial derivatives and to functions of more variables.

*Subjects:*
Mathematics.