The upper right picture shows the function y=(x), and the function is in x _ 0' (x _ 0) = lim {Δ x→ 0} [(x _ 0+Δ x)-(x _ 0)]/Δ x. If the function is differentiable in the continuous interval, the function has a derivative function in this interval, and it is recorded as' (x) or dy/.
Derivative definition
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.
The origin of the derivative words of this paragraph in folding editing
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 counting"; He called variable flow and the rate of change of variable flow, 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 to such a variable between these two different boundaries, 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.
Fold and edit this derivative function.
It is generally assumed that the unary function y=f(x) is defined in the neighborhood N(x0δ) of the point x0. When the increment of the independent variable is δ x = x-x0, the corresponding increment of the function is δ y = f (x0+δ x)-f (x0). If the limit of the ratio of function increment △y to independent variable increment △x exists and is limited when △x→0, the function f(x) is said to be derivable at x0, and this limit is called the derivative or rate of change of f at x0.
If the function f can be derived at every point in the interval I, a new function with I as its domain can be obtained, which is called f'(x) or y'. The derivative function called f cannot be simply called derivative.
Fold and edit the geometric meaning of this paragraph.
The geometric meaning of the derivative f'x0 of the function y=f(x) at x0 indicates the tangent slope of the function curve at P0 [the geometric meaning of x derivative 0fx0].
The geometric meaning of derivative is the tangent slope of function curve at this point.
Scientific application of folding editing this paragraph
Derivative is closely related to physical geometry algebra. Tangents can be found in geometry, instantaneous rate of change can be found in algebra, and velocity acceleration can be found in physics.
The concepts in derivative and wechat quotient differential are mathematical concepts abstracted from the problems of speed change and curve tangent, also called change rate.
For example, a car travels 600 kilometers in 10 hour, and its average speed is 60 kilometers per hour. However, in the actual driving process, the speed change is not always 60 km/h. In order to better reflect the speed change of the car during driving, the time interval can be shortened. Let the relationship between the position s of the car and the time t be: s=ft.
Then the average speed from time t0 to t 1 is:
[f(t 1)-f(t0)]/[t 1-t0]
When t 1 and t0 are infinitely close to zero, the speed of the car will not change much, and the instantaneous speed is approximately equal to the average speed.
Naturally, the limit lim [f (t/kloc-0 /→ t0)-f (t0)]/[t1-t0] is regarded as the instantaneous speed of the car at t0, which is the so-called speed. This is actually the process from average speed analogy to instantaneous speed. For example, the speed limit when we drive refers to the instantaneous speed.