Let the function y = f(x) be defined in the neighborhood of X, and both X and x+δ x are in this interval. If the function Δ y = f (x+Δ x)-the increment of f (x) can be expressed as Δ y = a Δ x+o (Δ x) (where a is a constant that does not change with Δ x, but a can change with x).

And o (Δ x) is an infinitesimal with a higher order than Δ x (note: o is pronounced as Omicron, Greek letter), then the function f o(δ x) is differentiable at point X, and Δ x is called the differential of the function corresponding to the dependent variable increment Δ y at point X, which is denoted as dy, that is, dy = a Δ x. The differential of the function is the main part of the function increment and is a linear function of Δ x, so the differential of the function is called the function increment.

Deduction:

Let the function y = f(x) be defined in a certain interval, and x0 and x0+△x are in this interval. If the function δδy = f(x0+δx)f(x0)f o (δ x0) can be expressed as δ y = a δ x+o (δ x), where a is a constant independent of δ x and o(δx) is δ. A Δ x is called the differential of the function corresponding to the increment of the independent variable Δ x at x0 point, and it is denoted as dy, that is, dy = a Δ x.

Differential dy is a linear function of the independent variable △ x, and the difference between dy and △y is a high-order infinitesimal quantity about △ X. We call dy the linear principal part of △ y .. It is concluded that △y≈dy when △x→0. The sign of the derivative is: (dy)/(dx)=f'(X).