Can anyone help me briefly explain the principles of calculus? And how to find a derivation? What is

The function of the sub-upper bound and its derivative.

Let the function f(x) be continuous over the interval [a,b], and let x be a point on [a,b]. Now let's look at the definite integral of f(x) over part of the interval [a,x], and we know that f(x) is still continuous on [a,x], so this definite integral exists.

If the upper limit x fluctuates arbitrarily over the interval [a,b], then for each given value of x, the definite integral has a corresponding value, so it defines a function on [a,b], denoted as (x):

Note: For clarity, we've changed the integral variable (definite integrals are not related to the notation of integral variables).

Theorem (1): If the function f(x) is continuous over the interval [a,b], then the function of the integral upper limit has a derivative on [a,b], and its derivative is (a x b).

2): If the function f(x) is continuous over the interval [a,b], then the function is a primitive function of f(x) on [a,b].

Note: Theorem (2) not only affirms the existence of the original function of continuous functions, but also preliminarily reveals the connection between the definite integral and the original function in integralism.

Newton-Leibniz formula.

Theorem (3): If the function f(x) is a primitive function of the continuous function f(x) over the interval [a,b], then.

Note: This formula is known as the Newton-Leibniz formula, and it further reveals the connection between the definite integral and the original function (indefinite integral).

It shows that the definite integral of a continuous function over the interval [a,b] is equal to the increment of any one of its original functions on [a,b]. So it is.

A given integral provides an efficient and easy way to calculate it.

In fact, it is a small amount analysis, which uses the limit method to analyze the rate of change of the function. I'm afraid that there are a lot of formulas that you have to be by yourself, of course, you can also push them yourself every time.

Refer to the Mathematical Analysis Book.

Everything you want is in the book.

Why not find a book.

The basics of calculus.

The study of functions, the study of the change of the motion of things in terms of quantity is the basic method of calculus. This method is called mathematical analysis.

Originally, in a broad sense, mathematical analysis included many sub-disciplines such as calculus and function theory, but now it is generally customary to equate mathematical analysis with calculus, and mathematical analysis has become synonymous with calculus. The basic concepts and content of calculus include differential calculus and integral calculus.

The main contents of differential calculus include: limit theory, derivatives, differentiation, etc.

The main contents of integral science include: definite integral, indefinite integral, etc.

Calculus was developed in connection with applications, and Newton originally applied calculus and differential equations to derive Kepler's three laws of planetary motion from the law of gravitation. Since then, calculus has greatly promoted the development of mathematics, as well as astronomy, mechanics, physics, chemistry, biology, engineering, economics and other branches of natural sciences, social sciences and applied sciences. And in these disciplines there are more and more widely used, especially the emergence of computers is more conducive to the continuous development of these applications.

The basic method of calculus derivation is to calculate the derivative shield kernel of a function according to the chain rule, that is, to find the derivative using the definitive formula of differentiation. First you need to find the expression of the function, and then use the definition of the differential to calculate the derivative of the function. In addition, the derivative can also be found using the extreme manuscript limit method, i.e., it is not necessary to find the expression of the function, but rather to find the derivative according to the limit case of the independent variable x.

If it is a complex multivariate function, you can use partial derivatives to find the derivatives of the respective variables to the value of the function.

The method is as follows, please comma circle for reference:

If there is help from the landslide, please celebrate.

The derivative of calculus is the process of determining the slope of a function. The derivative of the function can be obtained, and the derivative can be used to determine the minimum and maximum values of the function and the increasing and decreasing trend of the function. In the process of derivation, it is necessary to master a variety of derivative rules, including derivative rules, basic derivative formulas, gradients, total derivatives, etc.

At the same time, it is necessary to understand the physical and geometric significance of derivation. In the practice of imaginary calculations, it is also necessary to be proficient in the use of calculus software and mathematical tools. By constantly practicing and understanding the derivative knowledge and methods of calculus, it is possible to achieve excellent results in mathematics and engineering disciplines.

Differentiation in calculus is a common mathematical technique used to calculate the slope or rate of change of the slope of the burn of a sweatpants. In general, the derivative steps involve taking the implicit number lines of the parameters of a function to derive and using the chain rule to derive complex functions.

Derivative: Simply put, the slope somewhere in the function.

Differentiation: That is, to divide the function into infinitesimal parts, we divide the differential dy=f'(x)

dx, put f'(x) is seen as the slope k

This constitutes a function dy=f'(x)

dx, which is a primary function with an independent variable of dx, i.e. a function with a tangent somewhere in the function (not accurately). Because there's a b, this one is just an incremental function. ）

Integral: It's the original function.

Okay, let's sum it up, that is. The derivative is the slope of the tangent of the function, the differentiation is the function of the tangent of the function, and then the integral is the original function.

Integration is the inverse of a differentiation. Differentiation is equivalent to derivation.

Analysis: (1).f(x)=x 2+2x+c is an increasing function on [1,+.

Proposition p"x 1, x 2 + 2 x + c 7 2 constant"is a false proposition, i.e., f(x)=x 2+2x+c, the minimum value of f(1)<7 2 on [1,+, then 1+2+c<7 2

c<1/2.

2).x 2 is an increasing function on (0,1 2], and Evergrande is 0;

When c>1, log(c)x is always less than 0 at (0,1 2], which does not meet the requirements of the question.

When 0 so g(x)=x 2-log(c)x, is an increasing function on (0,1 2, then the proposition q: inequality x 2-log(c) x 0, at (0,1 2 is a true proposition.

Equivalent to the function g(x)=x2-log(c)x, the maximum value on (0,1 2 is less than or equal to 0.).

i.e. g(1 2) 0

i.e. 1 4-log(c)(1 2) 0

i.e. log(c)(1 2) 1 4=log(c)[c(1 4)] 0 c (1 4) 1 2

c 1 16, (take the fourth power on both sides at the same time.) )

1/16≤c<1.

In summary, 1 16 c< 1 2