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 increment of the function Δ y = f (x+Δ x)-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), o (Δ x) is infinitely smaller than the higher order of Δ x.

Usually, the increment Δ x of the independent variable X is called the differential of the independent variable, which is denoted as dx, that is, dx = Δ x. Then the differential of the function y = f(x) can be written as dy = f'(x)dx. The quotient of the differential of the dependent variable and the differential of the independent variable is equal to the derivative of the function, so the derivative is also called wechat quotient.

Dlnx and dx have different meanings:

1 and dlnx mean to integrate the whole lnx.

1 and dx represent the integral X.

Integral is the core concept in calculus and mathematical analysis. Usually divided into definite integral and indefinite integral. Intuitively speaking, for a given positive real function, the definite integral in the real number interval can be understood as the area value (a definite real value) of the curve trapezoid surrounded by curves, lines and axes on the coordinate plane.