Can anyone tell me what the role of derivatives is in industrial production?

Derivatives are the basic knowledge of higher mathematics, which is very extensive in theoretical research. I studied engineering, and a series of physical calculations, sensor calculations, control quantities, and especially signal calculations all require a solid knowledge of advanced mathematics.

The geometric meaning is to find the slope of the tangent. The physical meaning is that the velocity is obtained by the derivative of displacement, and the acceleration is obtained by the second derivative. The properties of the study functions include monotonicity, extremum, curve concave and convex, and inflection points. Use the derivative to find the maximum and minimum values of the function.

The most superficial term for derivative is a method (tool) for analyzing the law of change of functions, and a function is a method for analyzing the changes in the posture of everything in the world, that is, the derivative is the method (tool) for human beings to break the laws of nature.

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

Definition of derivative.

Let the function y=f(x) be defined at and near the point x=x0, and when the independent variable x has a change of x( x can be positive or negative) at x0 , then the function y has a corresponding change of y=f(x0 x) f(x0), and the ratio of these two changes is called the average rate of change of the function y=f(x) between x0 and x0 x.

If there is a limit when x 0, we say that the function y=f(x) is derivative at the point x0, and this limit is called the derivative of f(x) at the point x0 (i.e., the instantaneous rate of change), denoted as f(x0) or, i.e.

The derivative of the function f(x) at the point x0 is the limit of the mean rate of change of the function when the realignment of the independent variable tends to zero If the limit does not exist, we say that the function f(x) is not derivative at the point x0.

2. The method of finding the derivative.

Defined by the derivative, we can get the method of finding the derivative of the function f(x) at the point x0:

1) Find the increment of the function y=f(x0 x) f(x0);

2) find the average rate of change;

3) Take the limit and get the derivative.

3. The geometric meaning of derivatives.

The geometric meaning of the derivative of the function y=f(x) at the point x0 is the slope f (x0) of the tangent of the curve y=f(x) at the point p(x0,f(x0)).

Correspondingly, the tangent equation is y y0=

f′(x0)(x－x0).

4. Derivatives of several common functions.

The derivative of the function y=c (c is a constant).

c′=0.The derivative of the function y=xn(n q).

xn)′=nxn－1

The derivative of the function y=sinx.

sinx)′=cosx

Derivative of the function y=cosx.

cosx)′=sinx

5. Derivation of the four rules of function.

and the derivative. u＋v)′=u′＋v′

Bad derivative. u－v)′=

The derivative of the product of u v.

u·v)′=u′v＋uv′

Derivative of quotient. 6. Derivation of composite functions.

In general, the derivative y x of the composite function y=f[ (x)] on the independent variable x is equal to the derivative y u of the known function on the intermediate variable u= (x), multiplied by the derivative u x of the intermediate variable u on the independent variable x, i.e., y x = y u·u x

7. The logarithm of the manuscript letter and the derivative of the exponential function.

1) The derivative of the logarithmic function.

The formula cannot be entered.

where Eq. (1) is a special case of Eq. (2), and when A=E, Eq. (2) is Eq. (1).

2) The derivative of the exponential function.

ex)′=ex

ax)′=axlna

where Eq. (1) is a special case of Eq. (2), and when A=E, Eq. (2) is Eq. (1).

The derivative, also known as the microquotient, is the differentiation quotient of the dependent variable and the differentiation of the independent variable; Integrating the derivative gives you the original function (which is actually the sum of the original function and a constant).

Introduction: When learning functions, most people still learn unary functions, in which there is only one independent variable and one strain variable. But as people learn more and more advanced mathematics, they will find that there are also multivariate functions.

The independent variables of these functions are multi-liter grinding, so a new concept will emerge when finding the derivative, which is the partial derivative. The derivative method of partial derivative is not very different from that of ordinary derivative methods, that is, the derivative of a single independent variable is separately obtained, and a partial derivative is formed. Then judging the existence of partial derivatives is the first step in learning.

Years old. <> to determine whether the partial derivative exists and whether the function is continuous at this point, the most important thing is to look at the limit. For example, in a binary function, there is an independent variable, x, and for a certain value in the independent variable x, if the derivative limit of a tiny amount is added, then the partial derivative exists. The same is true for other independent variables, soUltimately, it depends on whether the limit exists or not, so as to determine whether the partial derivative exists

The existence of a partial derivative cannot be judged solely by whether the function is continuously biased at that point, which is very different from the previous unary functions.

In fact, there is not much difference between the partial derivative and the derivative learned before, but the derivative is found in a variety of situations. After finding the partial derivative, you can also do a quadratic derivative, soThe most important thing is the carefulness of the calculation, as long as you master the method of calculation and be careful enough, the partial derivative will not be solved wrong.

The study of advanced mathematics is not as easy as people think, nor is it as difficult as people think. The most important thing is to listen to the lectures in class, follow the teacher's ideas, and write homework carefully after class, then you will find that advanced mathematics is also very simple.

Derivatives are tools used to reflect the local properties of a function.

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 of the object.

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 from 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 integration are a pair of inverse operations, and they are both the most fundamental concepts in calculus.

The geometric meaning is to find the tangent slope, the physical meaning is to find the velocity derived from the displacement, and the acceleration is obtained from the second derivative. The properties of the study functions include monotonicity, extremum, curve concave and convex, and inflection points. Use the derivative to find the maximum and minimum values of the function.

Derivative, also known as epoch and microquotient, is a mathematical concept abstracted from the problem of velocity change and the tangent problem of curves. Also known as the rate of change. Derivatives are an important fundamental concept in calculus.

When there is a derivative of a function, it is said to be derivable or differentiable. The derivable function must be continuous. Discontinuous functions must not be derivative.

Derivatives are essentially a process of finding the limit, and the four rules of operation of derivatives are the same as the four rules of operation of the limit.

Applications of derivatives.

1 Monotonicity of functions.

2 Extrema of the function.

3 Find the extrema of the function.

4 The maximum value of the function.

Do you mean doing questions or practical applications?

If it is a practical application, it has a wide range. We all know that the collective meaning of differentiation lies in the slope, that is, the speed of change.

In the field of economics, derivatives are widely used in the derivation of economic formulas.

The same is true in the field of physics.

Mathematics is the foundation of the natural sciences.

You can find the slope, the increase and decrease interval, the maximum and the minimum value.

The second-order derivative function is the derivative of the first-order derivative of the Hui file, and the increase and decrease of the first-order derivative can be judged, and the second-order derivative value of the stationary point is 0 In the neighborhood centered on the stationary point (the point where the first-order derivative = 0) is in Tongbi, the first-order derivative increases monotonically, and the derivative value of the stationary point = 0 On both sides of the stationary point, the value of the first-order derivative left-right + the stationary point is defeated by the minimum value point of the original function.

Red is the original function, and black is the derivative).

The definition of a derivative and its practical application are as follows:

Definition of Derivative: A derivative is a local property of a function, and the derivative of a function at a certain point describes the rate of change of the function around that point. Not all functions have derivatives, and a function does not necessarily have derivatives at all points.

If a function is derivative at a certain point, it is said to be derivative at that point, otherwise it is called non-derivative. However, the derivable function must be continuous; Discontinuous functions must not be derivative.

Practical application of derivatives: Derivatives are used to analyze changes. Taking a primary function as an example, we know that the image of a primary function is a straight line, and in analytic geometry, a primary function is just a straight line with a slope in analytic geometry, and if you give a derivative of a function, you will get the slope.

The derivative is an important component of differential calculus, which is an important tool for studying the functional beam quality and curvilinear behavior, and is also an important method for solving some optimization problems in practical life. ** The method of using derivatives to solve problems related to materials, costs, profits and site selection in real life.

Calculation of derivatives:

Calculating the derivative of a known function can be calculated using the limit of the ratio of change as defined by the derivative. In practical calculations, most of the common analytic functions of spring jujube can be regarded as the result of the sum, difference, product, quotient or mutual compounding of some simple functions. As long as the derivatives of these simple functions are known, the derivatives of more complex functions can be deduced according to the derivative law of derivatives.