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The definition of derivative in advanced mathematics
Derivative, also called derivative function value. Also known as WeChat quotient, it is an important basic concept in calculus. When the independent variable x of the function y=f(x) generates an increment δ x at the point x0, if there is a limit a in the ratio of the increment δ y of the function output value to the increment δ x of the independent variable when δ x tends to 0, then A is the derivative at x0, which is denoted as f'(x0) or df(x0)/dx.

Derivative is the local property of function. The derivative of a function at a certain point describes the rate of change of the function near that point. If the independent variables and values of the function are real numbers, then the derivative of the function at a certain point is the tangent slope of the curve represented by the function at that point. The essence of derivative is the local linear approximation of function through the concept of limit. For example, in kinematics, the derivative of the displacement of an object with respect to time is the instantaneous velocity of the object.

Not all functions have derivatives, and a function does not necessarily have derivatives at all points. If the derivative of a function exists at a certain point, it is said to be derivative at this point, otherwise it is called non-derivative. However, the differentiable function must be continuous; Discontinuous functions must be non-differentiable.

For differentiable function f(x), x? F'(x) is also a function called the derivative function of f(x). The process of finding the derivative of a known function at a certain point or its derivative function is called derivative. Derivative is essentially a process of finding the limit, and the four algorithms of derivative also come from the four algorithms of limit. Conversely, the known derivative function can also reverse the original function, that is, indefinite integral. The basic theorem of calculus shows that finding the original function is equivalent to integral. Derivation and integration are a pair of reciprocal operations, both of which are the most basic concepts in calculus.

Chinese name

derivant

Foreign name

derivative

presenter

Newton, Leibniz

Show time

17th century

application area

Mathematics (calculus), physics

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Four standards of "parameter classification" of derivatives (including handouts)

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navigate by water/air

definition

formula

Properties of derivative sum function

Derivative species

App application

The development of history

origin

Around 1629, the French mathematician Fermat studied the method of tangent curve and extreme value of function; 1637 or so, he wrote a manuscript "the method of finding the maximum value and the minimum value". When doing tangent, he constructed the difference f(A+E)-f(A) and found that the factor e is what we call the derivative f'(A). [ 1]

develop

/kloc-the development of productivity in the 0/7th century promoted the development of natural science and technology. On the basis of predecessors' creative research, great mathematicians Newton and Leibniz began to study calculus systematically from different angles. Newton's calculus theory is called "flow number". He called the variable flow and the rate of change of the variable flow number, which is equivalent to what we call derivative. Newton's major works on "Flow Number Theory" include Finding the Area of Curved Polygon, Calculation Method Using Infinite Polynomial Equation, Flow Number Theory and Infinite Series. The essence of stream number theory is summarized as follows: his emphasis is on univariate function, not multivariate equation; It lies in the composition of the ratio of the change of independent variables to the change of functions; The most important thing is to determine the limit of this ratio when the change tends to zero. [ 1]

mature

1750, D'Alembert put forward a viewpoint about derivative in the "differential" entry written for the fourth edition of Encyclopedia published by French Academy of Sciences, which can be simply expressed by modern symbols.

In 1823, Cauchy defined the derivative in his Introduction to Infinitesimal Analysis: If the function y=f(x) is continuous between two given boundaries of the variable X, and we specify a value for such a variable contained between these two different boundaries, then this variable will get an infinitesimal increment. After 65438+60' s, Wilstras created ε-δ language, which re-expressed all kinds of limits in calculus.

The theoretical basis of calculus can be roughly divided into two parts. One is the theory of real infinity, that is, infinity is a concrete thing and a real existence; The other is the theory of latent infinity, which refers to a thought process, such as infinite approach.

As far as the history of mathematics is concerned, both theories have some truth. The real infinity took 150 years.