If the partial derivatives of a function to x and y both exist in a neighborhood of this point and are continuous at this point, then the function is differentiable at this point.

Let function y=f(x) and f(x) be defined in the domain of X. If the change of independent variable at point X is related to the corresponding change of function y=A×x+ο(x) (where A has nothing to do with X), then function f(x) is differentiable at point X, and Ax is called the differential of function f(x) at point X..

Function introduction

Function, a mathematical term. Its definition is usually divided into traditional definition and modern definition. The essence of these two functional definitions is the same, but the starting point of narrative concept is different. The traditional definition is from the perspective of movement change, and the modern definition is from the perspective of set and mapping.

The modern definition of a function is to give a number set A, assume that the element in it is X, apply the corresponding rule F to the element X in A, and record it as fx to get another number set B, assume that the element in B is Y, and the equivalent relationship between Y and X can be expressed as y=f(x). The concept of function consists of three elements: domain A, domain B and corresponding rule F..

The core is the correspondence rule F, which is the essential feature of functional relationship. Function was originally translated by Li, a mathematician of Qing Dynasty in China, in his book Algebra. He translated this way because "whoever believes in this variable is the function of that variable", that is, the function means that one quantity changes with another quantity, or that one quantity contains another quantity.

Given a set of non-empty numbers A, the corresponding law F is applied to A, denoted as f(A), and another set of numbers B is obtained, that is, B=f(A). Then this relationship is called functional relationship, or function for short.

In the process of a change, the amount of change is called a variable (in mathematics, the variable is X, and Y changes with the change of X value), and some numerical values do not change with the variable, so they are called constants.