High School Mathematics, Derivatives, Finding Masters, High Numbers Derivatives Solving

Infinite approach, in fact, is a matter of limits. What we call x>0, it's not actually a number, it's a function, and it can be less than any given positive number. It should be said that 28+ x is infinitely close to 28, but in fact, it can never be equal to 28, because x is never equal to 0.

But we don't have to stop thinking about this, although it can't be equal to 0, but if x is infinitely small, so small that we can completely ignore its existence, then can we think that 28+ x=28? To make it acceptable to you, I'm going to give you a very simple example:

an=1-(1 2) n It is the sum of the first n terms of the proportional sequence 1 2 n. If we let n tend to infinity, then (1 2) n tends to infinitesimus, so to speak, given an arbitrarily small number >0, I can always find n such that (1 2) n< then can we think that when n tends to infinity, an=1?You may say no, because you will say that (1 2) n is always greater than 0, how can 1 minus a number greater than zero equal to 1?

But I would say that every second you pass is 1 2 seconds, and then 1 4 seconds. After (1 2) n seconds, if you think that an is definitely less than 1, then it seems that you can't get past your second, but in fact? You're done, naturally.

Hence we can say that an=1, when n tends to infinity. It can also be said that when n tends to infinity, (1 2) n = 0. That's where the limit lies.

I think then you should be kind of comfortable with 28 + x = 28 Actually, from another point of view of strict proof, we look at the distance between two numbers, which is to see how much "gap" they have, of course, on the number line, we measure the distance between two numbers as an absolute value, then.

28+△x)-28|=|△x|= x and because x tends to 0, the difference between the two numbers tends to 0, i.e., in a sense, we can use 28 to equate 28+ x

Of course, from a geometric point of view, a curve, given one point, when another point slides on the curve, their line is the secant of the curve, but imagine what happens when the moving point is infinitely close to the given point from the left and right? Let's take it to the extreme, so that they coincide, which is the tangent of the curve at this fixed point (because there is only an intersection point), which is the derivative at this point.

I'm also a freshman in college, and I love math. My awareness of the limits is limited to this, and there is nothing to be expected to criticize. If you still can't accept it, just leave QQ and communicate more in the future.

In fact, in the process of derivative verification, you don't necessarily have to say that the infinite approach is equal to 0, you can completely think of it as a 0 such as the derivative of x 2 at 1, we can calculate it like this, f'(1)=《f(1+δx)-f(1)》/δx

1+δx)^2-1》/δx

2δx+δx^2》/δx

2+δx=2

All steps should be preceded by a δx sign that tends to 0, and if you can't hit it, you won't play it) In fact, most of the δx in derivative proofs in high school mathematics can be reduced in the end, and you don't need to think about it at all....

Also, if conditions permit, you can find this mathematical analysis to take a look, which mainly says that the proof is more detailed....

Strict definitions are in college, and there's a special language for it, and it's useless for you to learn it.

Let the point be any point adjacent to the point, then the slope of the secant is , when , the point tends to the point and the inclination of the secant tends to the inclination of the tangent , then the slope of the tangent is .

The derivative formula for common functions of high numbers is shown in the figure below

Derivative is a mathematical method of calculation, which is defined as the limit of the quotient between the increment of the dependent variable and the increment of the independent variable when the DAO increment of the independent variable tends to zero.

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.

Extended information: The first derivative represents the rate of change of the function, and the most intuitive manifestation lies in the monotonicity of the function, theorem: let f(x) be continuous on [a,b], and have a first derivative in (a,b), then:

1) If in (a, b) f'(x) >0, then f(x) on [a,b] is monotonically increasing;

2) If f'(x)<0 in (a,b), then the graph of f(x) on [a,b] decreases monotonically;

3) If in (a, b) f'(x)=0, then the graph of f(x) on [a,b] is a straight line parallel (or coincidental) to the x-axis, i.e., constant on [a,b].

The derivative of a function is the slope of the tangent at a point. When the function increases monotonically, the slope is positive, and when the function decreases monotonically, the slope is negative.

Derivatives and Differentiation: Differentiation is also a way of linearly describing the variation of a function around a point. Differentiation and derivative are two different concepts. However, differentiability and derivability are exactly equivalent to unary functions.

A differentiable function whose differentiation is equal to the derivative multiplied by the differential dx of the independent variable, in other words, the differentiation quotient of the function and the differentiation of the independent variable is equal to the derivative of the function. Therefore, the derivative is also called a micro-quotient. The differentiation of the function y=f(x) can be denoted as dy=f'(x)dx。

Derivative is an important fundamental concept in calculus and is a local property of functions. 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. 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 not be derivative. The development of productive forces in the 17th century promoted the development of natural science and technology, and on the basis of the creative research of their predecessors, the great mathematicians Newton and Leibniz began to systematically study calculus from different perspectives.

Newton's theory of calculus is called "flow number technique", he called the variable flow, and the rate of change of the variable is the flow number, which is equivalent to what we call the derivative. Newton's main works on "flow number" are "Finding the Area of a Curvy Shape", "Calculation of Infinite Multinomial Equations" and "Flow Number and Infinite Series", and the essence of flow number theory is summarized as follows: his focus is on the function of one variable rather than on the equation of multiple variables; lies in the composition of the ratio of the change in the independent variable to the change in the function; The most important thing is to determine the limit of this ratio when the change is close to zero.

If the function y=f(x) is derivable at every point in the open interval, the function f(x) is said to be derivable in the interval. At this time, the function y=f(x) corresponds to a definite derivative value for each definite x value in the interval, which constitutes a new function, which is called the derivative of the original function y=f(x), denoted as y'、f'(x), dy dx, or df(x) dx, referred to as the derivative.

Derivatives are an important pillar of calculus. Newton and Leibniz contributed to this.

How it works:

Definitional Method. To find a derivative using the definition of a derivative, the following is an example of how to define it.

Formula method. To find the derivative according to the formula in the book, the following is an example problem about the formula method.

Composite function method.

To find the derivative using composite functions, the following is an example of the composite function method.

Implicit function method. Using implicit functions to find derivatives, the following is an example of the implicit function method.

Logarithmic method. The logarithmic method is suitable for finding derivatives of power functions and the product of the given function that can be regarded as a power, which can simplify the operation. The following is an example of the logarithmic method.

Piecewise function method.

The piecewise function is derived at the piecewise point. The following is an example of the law of indefiniteness.

Common derivative formulas:

c'=0 (c is a constant function);

x^n)'=nx^(n-1) (n∈q*)；

sinx)' cosx；

cosx)' sinx；

tanx)'=1/(cosx)^2=(secx)^2=1+(tanx)^2

cotx)'=1/(sinx)^2=(cscx)^2=1+(cotx)^2

secx)'=tanx·secx

cscx)'=cotx·cscx

sinhx)'=hcoshx

coshx)'=hsinhx

tanhx)'=1/(coshx)^2=(sechx)^2

coth)'=1/(sinhx)^2=-(cschx)^2

sechx)'=tanhx·sechx

cschx)'=cothx·cschx

e^x)' e^x；

a^x)'a xlna (ln is the natural logarithm).

inx)'1 x (ln is the natural logarithm).

logax)'xlna) (1), (a>0 and a is not equal to 1) (x 1 2).'=2(x^1/2)]^1)

1/x)'=x^(-2)

The other is the derivative of the composite function

u±v)'=u'±v'

uv)'=u'v+uv'

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

These latter high schools are not used, but they can be written directly when they encounter more mastery points, and they do not need to be converted into common functions to solve, arcsinx).'=1/(1-x^2)^1/2

arccosx)'=1/(1-x^2)^1/2

arctanx)'=1/(1+x^2)

arccotx)'=1/(1+x^2)

arcsecx)'=1/(|x|(x^2-1)^1/2)

arccscx)'=1/(|x|(x^2-1)^1/2)

The formula of the product sum and the difference and the difference product are used. The formula is as follows:

sin sin =-cos( +cos( -cos cos =[cos( +cos( -sin cos =[sin( +sin( -cos sin =[sin( +sin( -You do it yourself according to the formula, cosx is the same.

That's using the trigonometric and differential product formulas, i.e. .

sin(x+h)-sinⅹ

2cos(x+h+x/2)sin(x+h-x)/22cos(ⅹ+h/2)sinh/2

cos(x+h)sin(h/2)/(1/2)。

(1-lnx)/x^2=x^2-2ex+a。

Order: t(x)=x 2-2ex+a. This can be seen as the fact that the two functions h(x) and t(x) are equal when x takes one of the values.

In order to conclude that when x=e, h(x) obtains the maximum value and t(x) obtains the minimum value. Therefore, if the maximum value of these two functions is smaller than the minimum value, it means that the equation is unsolved; If it is greater than the minimum value, it means that there may still be an x value that is not equal to e, so that the two functions still have the same function value, that is, the real number is not unique; If it is equal to the minimum value, it means that h(x)=t(x) will only be made if x=e.

One idea is that the two sides of the equation can be regarded as two different functions, and when x takes a certain value, the two functions are equal in value.

For example, if the upper semicircle f(x) = (4-x 2) has an intersection with the straight line g(x)=kx+1 and the number of intersection points, it is actually the number of real solutions to the equation f(x)=g(x). (This problem can be solved by combining numbers and shapes, or it can be directly reduced to a one-dimensional quadratic equation and solved according to the discriminant formula.) ）

Because at x=e, the maximum is equal to the maximum, so there is only one root.

Derivation is calculated according to the composite function. For example, the second = cos[cos (tan2x)]*2cos(tan3x)))*sin(tan3x)(1 cos x) 3

Solution: Let the bottom of this triangle be x meters and the height be y meters.

x+1)y/2=xy/2+

x（y+1）/2=xy/2+

x=5 y=3

Area: 5 3 2 = square meters).