What Is a Flow Derivative?

Derivative, also called derivative function value. Also known as differential quotient, it is an important basic concept in calculus. When the independent variable x of the function y = f (x) produces an increase x at a point x ₀ , the ratio of the increase of the output value of the function y to the independent variable increase x is a limit a when x approaches 0, if it exists , A is the derivative at x ₀ , written as f '(x ₀ ) or df (x ₀ ) / dx.

Assume

Derivatives are closely related to physics, geometry, and algebra: you can find tangents in geometry; you can find the instantaneous rate of change in algebra; you can find speed and acceleration in physics.

Derivatives are also known as epochs and differential quotients (concepts in differentiation). They are mathematical concepts abstracted from the problem of speed change and the tangent of a curve (the direction of vector speed), also known as the rate of change.

For example, if a car travels 600 kilometers in 10 hours, its average speed is 60 kilometers per hour. However, during actual driving, there are changes in speed, not all of which are 60 km / h. In order to better reflect the speed change of the car during driving, the time interval can be shortened, and the relationship between the position s of the car and the time t is:

Then the average speed of the car from time t ₀ to t ₁ is:

When t ₁ approaches infinity infinitely, the speed of the car will not change greatly, and the average speed will be approximately equal to the instantaneous speed at time t _0. Therefore, the limit at this time is taken as the instantaneous speed of the car at time t ₀ , which is

This is what is commonly called speed. This is actually a process from the average speed analogy to the instantaneous speed (such as the speed limit when we drive refers to the instantaneous speed).

Derivative another definition: when x = x ₀ , f '(x ₀ ) is a certain number. In this way, when x changes, f '(x) is a function of x, we call it f (x) (derivative function about x), or derivative for short.

Some important concepts in physics, geometry, economics and other disciplines can be represented by derivatives. For example, the derivative can represent the instantaneous speed and acceleration of a moving object (for linear motion, the first derivative of displacement with respect to time is instantaneous velocity, and the second derivative is acceleration), which can indicate the slope of a curve at one point, and it can also represent economics. Margins and elasticity.

The above-mentioned definition of the classical derivative can be considered as reflecting the change of the function in the local Euclidean space. In order to study the changes of vector bundle cross sections (such as tangent vector fields) on more general manifolds, the concept of derivatives is generalized as so-called "connections". With the connection, people can study a wide range of geometric problems, which is one of the most important basic concepts in differential geometry and physics.

note:

1. f '(x) <0 is a necessary and sufficient condition that f (x) is a subtraction function, not a necessary and sufficient condition.

2. The point where the derivative is zero is not necessarily the extreme point. When the function is a constant value function, there is no increase or decrease, that is, there is no extreme point. But the derivative is zero. (A point with a derivative of zero is called a stagnation point. If the signs of the derivatives on both sides of the stagnation point are opposite, the point is an extreme point, otherwise it is a general stagnation point, such as

In f '(0) = 0, the sign of the left and right derivatives of x = 0 is positive, and this point is a general stagnation point. )

Derivative method (definition method):

Find the increment of the function

;

Find the average rate of change;

Take the limit and get the derivative.