What is the derivative as the title Thank you

The derivative itself is an ordinary function, there are independent variables and function values, and the derivative value is the function value. It's just that the functional relation of the derivative function is obtained by the special method of finding the limit of another function, that is, the original function, so why invent this thing? For example, the car does variable speed linear motion on the road, one ha fast and one ha slow, the displacement is a function of time, and the mathematical expression is to search for s=f(t), and the speed at a certain moment is required, what should I do?

It is possible to remove the interval in a certain period of time, i.e. v= s t, but this is the average velocity, which is not precise enough, to get the instant velocity, you can use the limit method: if t tends to 0, then v= s t tends to a definite limit value, and this value is the exact immediate velocity. For any time t, we get a function:

velocity v=f(t), this function is the derivative of s=f(t). The problem of finding the rate of change (i.e., how fast the value of a function changes with the change of the independent variable) can be obtained by calculating the derivative. Let the original function be y=f(x), obviously y changes with x, when the amount of x change is x, the corresponding y has a change y, the ratio, y x reflects the speed of y change with x, is an average value, to get the speed of change at x point, it is required to find the limit value of y x when x 0, and this limit value is the derivative of the original function y=f(x).

To describe it mathematically, it should be simple and precise, so use "If when x, y x has a limit, we say that the function y=f(x) is at the point x." and call this limit f(x) at the point x. (or rate of change)" to describe the meaning of the derivative.

The derivative of the graph of the function at a point is the slope of the change point.

Derivative is short for derivative function.

To put it bluntly, the derivative is actually the slope.

Find the increment of the function δy=f(x0+δx)-f(x0) to find the average rate of change.

Take the limit, and the side key gives the derivative. Derivative formulas for common functions:

c'=0 (c is constant);

x^n)'=nx^(n-1) (n∈q)；

sinx)'=cosx；

cosx)'=sinx；

e^x)'=e^x；

a^x)'=a xina (Imitation of ln is the natural logarithm) The four rules of operation for derivatives:

Big do (u v).'=u'±v'

uv)'=u'v+uv'

u/v)'=u'v-uv')/v^2

Derivatives are local properties of functions. The derivative of a function at a point describes the rate of change of the function around that point. If both the independent variables and the values of the function are real, the derivative of the function at a point is the tangent slope of the curve represented by the function at that point.

The essence of derivatives is to perform a local linear approximation of a function through the concept of limits. For example, in kinematics, the derivative of the displacement of an object with respect to time is the instantaneous velocity rise of the object.

Not all functions have derivatives, and a function does not necessarily have derivatives at all points. If a function exists at a certain point in derivative, it is said to be derivable at that point, otherwise it is called underivable. However, the derivable function must be continuous; Discontinuous functions must be delicate and should not be noisy and vertical.

The process of finding the derivative of a known function at a point or its derivative is called derivative. In essence, derivation is a process of finding the limit, and the four rules of operation of derivatives are also the same as the four rules of operation of the limit. Conversely, a known derivative can also be reversed to find the original function, i.e., an indefinite integral.

The fundamental theorem of calculus states that the original function is equivalent to the integral. Derivative and integral are a pair of reciprocal operations, and they are both the most fundamental concepts in calculus.

Semanticly, the reason why a derivative is called a derivative is that it is derived from a function. Etymologically, the derivative of the Chinese term ** is derived from the English derivative, and the literature shows that the English derivative is ** from the French word fonction dérivée, which was first named by the French mathematician lagrange.

Derivatives and derivatives should not be generalized.

derivative function

Derivative: differential coefficient

Before liberation and in the early days of liberation, derivatives were not called derivatives, but micro-quotients, that is, micro-quotients; The derivative is a name that was later recalled.

Because the derivative is the instantaneous rate of change of a function. If the derivative is 0, it indicates that the value of the function is increasing; If the derivative < 0, the value of the function is decreasing; Therefore, as the name suggests, the derivative has the ability to guide or guide the changing trend of the function, so it is also named derivative. Therefore, the derivative can better reflect the essential properties of the function than the microquotient.

The most superficial way of saying derivative is a method (tool) for analyzing the law of change of functions, and functions are a method for analyzing the changes of everything in the world before encountering forests, that is, derivatives are the methods (tools) for human beings to break the laws of nature.

The meaning of derivatives in different fields has different interpretations, Xiaoqing in mathematical functions it represents the slope; In the relationship between physical displacement and time, it is instantaneous velocity, acceleration; In economics, the derivative can be used to analyze actual dynamic changes, for example, it can represent the marginal cost of spring rulers. This is also the role of the derivative in practical applications, and anything that changes, through the derivative, its transient can be analyzed.

Summary. Derivative, also known as derivative value. Also known as micro-quotient, it is an important basic concept in calculus.

When the independent variable x of the function y=f(x) produces an incremental δx at a point x0, the ratio of the incremental δy of the output value of the function to the incremental δx of the independent variable is at the limit a when δx approaches 0 if it exists, a is the derivative at x0 and is denoted as f'(x0) or df(x0) dx.

The derivative refers to derivative, also known as the value of the lead. Also known as micro-quotient, it is an important basic concept in calculus. When the independent variable x of the function y=f(x) produces an incremental δx at a point x0, the ratio of the output value of the function to the increment δx of the independent variable is at the limit a when δx approaches 0, and if it exists, a is the derivative at x0 and is denoted as f'(x0) or df(x0) dx.

Do derivatives help for initial numbers.

I'm now in my third year of junior high school, and I want to see the derivatives.

If you are helpful, you can learn it in advance.

Meaning: The original meaning of the derivative is "difference", and the English symbol d

The mathematical meaning of a derivative is the ratio of the amount of change of two variables; The geometric meaning is the slope of a point on a curve.

Purpose:1The monotonic interval of the judgment function: d>0, monotonically increasing; d<0, monotonically decreasing;

2.Judge the shape of the curve: the second-order guide is less than or equal to 0, convex; The second-order conductance is greater than or equal to 0 concave;

3.Find the extrema and the maximum: the first derivative d=0, which may be the extreme point; At the same time, the second derivative is 0, the minimum point;

At the same time, the second derivative <0, which is the maximum point;

Is it okay upstairs? The question is about the derivative of the derivative, not the derivative.

The derivative is the slope of the tangent of the curve at a certain point. There are many functions, simply put, it is to replace the curve with a straight line at this point, and transform the nonlinear problem into a linear problem.

It is generally used to determine whether the curve is convex or concave.

The second-order conductance of the convex type is less than or equal to 0

The second-order conductor of the concave type is greater than or equal to 0