First, the first definition of derivative.
Let the function y = f(x) be defined in the neighborhood of point x0. When the independent variable x has an increment △x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets an increment △y = f(x0+△x)-f(x0). If the ratio of △y to △x exists when △ x→0, the function y = is called.
Second, the second definition of derivative
Let the function y = f(x) be defined in the neighborhood of point x0. When the independent variable x changes △x at △x (x-x0 is also in the neighborhood), the function changes △y = f(x)-f(x0) accordingly. When △x→0, if there is a limit in the ratio of △y to △ x, the function y = f(x) exists.
Third, derivative function and derivative
If the function y = f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y = f(x) corresponds to a certain derivative of each certain value of x in the interval I, and constitutes a new function. The derivative functions of the original function y = f(x) are called y', f' (x), dy/dx and df (x)/dx. Derivative function is called derivative for short.
Extended data
First, the early derivative concept-special form
About 1629, the French mathematician Fermat studied the method of making the tangent of the curve and finding the extreme value of the function. 1637 or so, he wrote a manuscript to find the method of maximum and minimum. He constructed the difference f(A+E)-f(A) when tangent, and found that the factor e is what we call the derivative f'(A).
Two. 17th century-widely used "flow counting"
/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 main works on "flow counting" are "Finding the area of a curved polygon", "Calculation with infinite polynomial equation" and "Flow counting and infinite series". The essence of the flow counting theory is that its focus lies in the function of a variable, not in the multivariable equation, but in the composition of the ratio of the change of the independent variable to the change of the function, which is the limit that determines this ratio when the change tends to zero.
Three. /kloc-derivative of the 0/9th century-gradually mature theory
1750, D'Alembert put forward a viewpoint about derivative in the "differential" entry written for the fifth edition of Encyclopedia published by French Academy of Sciences, which can be simply expressed by the modern symbol {dy/dx)=lim(oy/ox).
1823, Cauchy defined the derivative in Introduction to Infinitesimal Analysis. If the function y=f(x) is continuous between two given boundaries of the variable X, and we assign a value between these two different boundaries for such a variable, then this variable will get an infinitesimal increment.
After 65438+ 1960' s, Wilstrass created ε-δ language, and the definitions of derivatives of various limit superposition expressions in calculus obtained the common forms today.
4. Real infinity will make the sudden second round of calculus elementary or possible. 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, and the other is latent infinity, which refers to an ideological process such as infinite approximation.
Historically, both theories have some truth. It takes 150 years of real infinity, and then uses the current limit theory.
Whether light is electromagnetic wave or particle is a long-debated problem in physics, which was later unified by wave-particle duality. Calculus is not the best method, whether using modern limit theory or the theory of 150 years ago.
References:
Derived Baidu Encyclopedia Partial Derivative Baidu Encyclopedia