There are some differences between derivative and differential in writing, for example, y'=f(x) is derivative and dy=f(x)dx is differential. Integration is to find the original function, which can be understood as the inverse operation of the derivative of the function.

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, and its derivative is y'=f'(x).

Let f(x) be an primitive function of function f(x), and we call all primitive functions F(x)+c (c is an arbitrary constant) of function f (x) indefinite integrals. The mathematical expression is: if f'(x)=g(x), there is ∫ g (x).

Extended data:

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 independent of Δ x), o (Δ x) is an infinitesimal higher than Δ x (note: o is pronounced as Omicron, Greek letter).

Then the function f(x) is differentiable at point X, and aδX is called the differential of the function corresponding to the dependent variable increment δy at point X, which is denoted as dy, that is, dy = aδX X 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 the linear main part of the function increment (δ x→ 0).

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 of a function is equal to the derivative of the function. Therefore, the derivative is also called WeChat business.

When the independent variable x becomes X+△X, the function value changes from f(X) to f(X+△X) accordingly. If there is a constant A that has nothing to do with △X, so that the difference between f(X+△X)-f(X) and A △X is the higher order infinitesimal of △X→0 about △X, it is called A △. In unary calculus, differentiability and differentiability are equivalent. Write a △ x = dy, then dy = f ′ (x) dx. For example: d(sinX)=cosXdX.

The concept of differentiation is produced in solving the contradiction between straight and curved. In tiny parts, straight lines can be used instead of curves to approximate, and its direct application is the linearization of functions. Differential has a double meaning: it represents a tiny quantity, so the numerical calculation result of a linear function can be regarded as the numerical approximation of the original function, which is the basic idea of approximate calculation by differential method.

The driving force of overall development comes from the demand in practical application. In practice, some unknowns can sometimes be roughly estimated, but with the development of science and technology, it is often necessary to know the exact values. If the area or volume of simple geometry is needed, the known formula can be applied. For example, the volume of a rectangular swimming pool can be calculated by length x width x height.

But if the swimming pool is oval, parabolic or more irregular, it is necessary to calculate the volume by integral. In physics, it is often necessary to know the cumulative effect of one physical quantity (such as displacement) on another physical quantity (such as force), and integration is also needed at this time.

The emergence of Lebesgue integral stems from the need to deal with more irregular functions in probability theory and other theories Riemann integral can't deal with the integral problem of these functions. So we need a broader concept of integral, so that more functions can define integral. At the same time, the definition of new integral should not conflict with Riemannian integrable function. Leberg integral is such an integral.

Riemann integral defines the concept of integral for elementary function and piecewise continuous function, while Lebesgue integral extends the definition of integral to measure space.

The concept of Lebesgue integral is defined on the concept of measure. Measurement is the generalization of measuring length and area in daily concepts, and it is defined in an axiomatic way. Riemann integral can actually be regarded as a series of rectangles covering the graph below the function curve as much as possible. The area of each rectangle is the length times the width, or the product of the lengths of two intervals.

Measure defines the concept of similar length for a set in a more general space, so that the area of a graph can be "measured" under a more irregular function curve, thus defining an integral. In one-dimensional real space, the Lebesgue measure μ(A) of an interval A= [a, b] is the right end value minus the left end value of the interval, b? Answer: This makes Lebesgue integral compatible with Riemann integral in the normal sense.

In a more complicated situation, the integral set can be more complicated. It is no longer an interval, or even the intersection or union of intervals, and its "length" is given by the measure.

References:

Baidu Encyclopedia-Differential? Baidu encyclopedia-integral