How is the derivative produced? Please tell us about the economic background and mathematical background of the derivative!

Calculus is a branch of mathematics that studies the differentiation and integration of functions and related concepts and applications.

It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the calculation of derivative, is a set of theories about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of universal symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume.

The basic theorem of calculus points out that differential and integral are reciprocal operations, which is why the two theories are unified into calculus. We can discuss calculus from either of them, but in teaching, differential calculus is usually introduced first.

Calculus is a general term for differential calculus and integral calculus. It is a mathematical idea, in which' infinite subdivision' is differential and' infinite summation' is integral. Infinity is the limit, and the idea of limit is the basis of calculus, which is to look at problems with a moving idea. For example, the instantaneous speed of a bullet flying out of a gun bore is the concept of differentiation, and the sum of the distances traveled by a bullet at each moment is the concept of integration. If the whole mathematics is compared to a big tree, then elementary mathematics is the root of the tree, and various branches of mathematics are branches, while the main part of the trunk is calculus. Calculus is one of the greatest achievements of human wisdom.

the concepts of limit and calculus can be traced back to ancient times. In the second half of the seventeenth century, Newton and Leibniz completed the preparatory work that many mathematicians had participated in, and established calculus independently. Their starting point of establishing calculus is intuitive infinitesimal, and their theoretical foundation is not solid. It was not until the 19th century that Cauchy and Wilstrass established the limit theory, and Cantor and others established the strict real number theory, so that the discipline could be rigorous.

Calculus is developed in connection with practical application, and it is more and more widely used in many branches such as astronomy, mechanics, chemistry, biology, engineering, economics and other natural sciences, social sciences and applied sciences. In particular, the invention of computers is more conducive to the continuous development of these applications.

Everything in the objective world, from particles to the universe, is always moving and changing. Therefore, after introducing the concept of variable in mathematics, it is possible to describe the movement phenomenon in mathematics.

due to the generation and application of the concept of function and the need of the development of science and technology, a new branch of mathematics came into being after analytic geometry, which is calculus. Calculus plays a very important role in the development of mathematics. It can be said that it is the biggest creation in all mathematics after Euclidean geometry.

The establishment of calculus

In terms of calculus becoming a discipline, it was in the 17th century, but the ideas of differential and integral had already come into being in ancient times.

in the 3rd century BC, Archimedes of ancient Greece implied the idea of modern integral calculus when he studied and solved the problems of parabolic arch area, sphere and spherical cap area, area under spiral and volume of rotating hyperbola. As the basis of differential calculus, the limit theory was clearly discussed as early as ancient times. For example, in the book "Zhuangzi" written by Zhuang Zhou in our country, it is recorded that "a foot's space is inexhaustible." Liu Hui in the Three Kingdoms period mentioned in his circle cutting that "if you cut it carefully, you will lose little, and if you cut it again, you will lose nothing with the circumference and body." These are simple and typical limit concepts.

in the 17th century, there were many scientific problems to be solved, and these problems became the factors that prompted calculus. To sum up, there are about four main types of problems: the first type is the problem that appears directly when studying sports, that is, the problem of seeking instant speed. The second kind of problem is to find the tangent of the curve. The third kind of problem is to find the maximum and minimum of a function. The fourth problem is to find the length of the curve, the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of the object, and the gravity of one object with a considerable volume acting on another object.

Many famous mathematicians, astronomers and physicists in the 17th century have done a lot of research work to solve the above problems, such as Fermat, Descartes, Robois and Girard Desargues. Barrow and Varys in Britain; Kepler in Germany; Italian Cavalieri and others have put forward many fruitful theories. Contributed to the creation of calculus.

In the second half of the 17th century, on the basis of predecessors' work, Newton, a great British scientist, and Leibniz, a German mathematician, independently studied and completed the creation of calculus in their own countries, although this was only a very preliminary work. Their greatest achievement is to connect two seemingly unrelated problems, one is the tangent problem (the central problem of differential calculus) and the other is the quadrature problem (the central problem of integral calculus).

Newton and Leibniz established calculus from the intuitive infinitesimal, so this subject was also called infinitesimal analysis in the early days, which is the source of the name of the big branch of mathematics now. Newton's study of calculus focused on kinematics, while Leibniz focused on geometry.

Newton wrote "Flow Method and Infinite Series" in 1671, which was not published until 1736. In this book, he pointed out that variables are produced by the continuous movement of points, lines and surfaces, which denied that variables he thought before were static sets of infinitesimal elements. He called continuous variables flow quantities and the derivatives of these flow quantities flow numbers. Newton's central problems in stream number technique are: knowing the path of continuous motion, finding the speed at a given moment (differential method); Given the speed of motion, find the distance traveled in a given time (integral method).

Leibniz of Germany is a knowledgeable scholar. In 1684, he published what is considered to be the earliest calculus document in the world. This article has a very long and strange name: A New Method for Finding Maximal Minimax and Tangent, which is also applicable to Fractions and Irrational Quantities, and the wonderful type of calculation of this new method. It is such an article with vague reasoning, but it has epoch-making significance. He is famous for containing modern differential symbols and basic differential laws. In 1686, Leibniz published the first literature on integral calculus. He is one of the greatest semiotics scholars in history, and the symbols he created are far superior to Newton's symbols, which has a great influence on the development of calculus. The universal symbol of calculus that we use now was carefully selected by Leibniz at that time.

The establishment of calculus has greatly promoted the development of mathematics. In the past, many problems that elementary mathematics was helpless were often solved by using calculus, which showed the extraordinary power of calculus.

As mentioned earlier, the establishment of a science is by no means the achievement of one person. It must have been completed by one person or several people after the efforts of many people and the accumulation of a large number of achievements. So is calculus.

Unfortunately, while people appreciate the magnificent function of calculus, when they put forward who is the founder of this subject, it actually caused an uproar, which caused a long-term confrontation between mathematicians in continental Europe and British mathematicians. British mathematics was closed to the outside world for a period of time, limited by national prejudice, and stuck too rigidly in Newton's "flow counting", so the development of mathematics fell behind for a whole hundred years.

In fact, Newton and Leibniz studied independently, respectively, and completed them in roughly the same time. What is more special is that Newton founded calculus about 1 years earlier than Leibniz, but Leibniz published the theory of calculus three years earlier than Newton. Their research has both advantages and disadvantages. At that time, due to national prejudice, the debate about the priority of invention lasted for more than 1 years from 1699.

It should be pointed out that this is the same as the completion of any major theory in history, and the work of Newton and Leibniz is also very imperfect. They have different opinions on the question of infinity and infinitesimal, which is very vague. Newton's infinitesimal, sometimes zero, sometimes not zero but a finite small amount; Leibniz's can't justify himself. These basic defects eventually led to the second mathematical crisis.

Until the early 19th century, the scientists of French Academy of Sciences, led by Cauchy, made a serious study of the theory of calculus and established the limit theory. Later, it was further tightened by the German mathematician Wilstrass, making the limit theory a firm foundation of calculus. Only then can calculus be further developed.

any emerging and promising scientific achievement attracts the vast number of scientific workers. In the history of calculus, there are also some stars: Jacques Bernoulli of Switzerland and his brothers John Bernoulli, Euler, Lagrange of France, Cauchy ...

Euclidean geometry and algebra in ancient and medieval times are all constant mathematics, and calculus is the real variable mathematics, which is a great revolution in mathematics. Calculus is the main branch of higher mathematics, which is not limited to solving the problem of variable speed in mechanics. It gallops in the garden of modern science and technology and has made countless great achievements.

The basic content of calculus

It is the basic method of calculus to study the function and the movement and change of things from the quantitative aspect. This method is called mathematical analysis.

Originally, in a broad sense, mathematical analysis included calculus, function theory and many other branches, but now it is generally used to equate mathematical analysis with calculus, and mathematical analysis has become a synonym for calculus. When mathematical analysis is mentioned, it is known that it refers to calculus. The basic concepts and contents of calculus include differential calculus and integral calculus.

The main contents of differential calculus include: limit theory, derivative, differential and so on.

The main contents of integral include definite integral, indefinite integral and so on.

Calculus is developed in connection with scientific application. At first, Newton used calculus and differential equations to analyze Tycho's vast astronomical observation data, obtained the law of universal gravitation, and further derived Kepler's three laws of planetary motion. Since then, calculus has become a powerful engine to promote the development of modern mathematics, and it has also greatly promoted the development of various branches of natural science, social science and applied science such as astronomy, physics, chemistry, biology, engineering and economics. And it is widely used in these disciplines, especially the appearance of computers is more conducive to the continuous development of these applications.

definition of unary differential

: let the function y = f(x) be defined in a certain interval, and x and x+δ x are in this interval. If the increment of the function Δy = f(x+Δx)? F(x) can be expressed as δ y = a δ x+o (δ x) (where a is a constant independent of δ x), and o (δ x) is infinitely smaller than δ x, then the function f(x) is said to be differentiable at point x, and a δ x is called the differential of the function at point x corresponding to the increment δ x of the independent variable, which is denoted as dy, that is, dy =

the increment Δx of the independent variable x is usually called the differential of the independent variable, which is denoted as dx, that is, dx = Δx x. Then the differential of the function y = f(x) can be written as dy = f'(x)dx. The quotient of the differential of a function and the differential of an independent variable is equal to the derivative of the function. Therefore, the derivative is also called Wechat business.

geometric meaning

let Δx be the increment of point m on the curve y = f(x) on the abscissa, Δy be the increment of curve at point m corresponding to Δx on the ordinate, and dy be the increment of curve tangent at point m corresponding to Δx on the ordinate. When |Δx| is very small, |Δy-dy | is much smaller than |Δy－dy| (high-order infinitesimal), so near point m, we can approximate the curve segment with a tangent line segment.

[ Edit this paragraph] Multiple differential

Similarly, when there are multiple independent variables, the definition of multiple differential can be obtained.

integration is the inverse operation of differentiation, that is, knowing the derivative function of a function, and finding the original function in reverse. In application, the function of integration is not only that, but it is widely used in summation, which is to find the area of curved triangle. This ingenious solution method is determined by the special properties of integration.

the indefinite integral of a function (also called the original function) refers to another family of functions, and the derivative function of this family of functions is just the previous function.

where: [F(x)+C]' = f(x)

The definite integral of a real variable function in the interval [a,b] is a real number. It is equal to the value in B minus the value in A of an original function of this function.

the differential corresponding to the first derivative of the first-order differential

function is called the first-order differential;

the first-order differential is called the second-order differential;

......

The differential of the n-order differential is called the (n+1)-order differential

, that is, d(n)y = f (n) (x) * dx^n (f (n) (x) refers to the n-order derivative, fans d (n) y refers to the n-order differential, dx.

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