First of all, the method of derivation
(1) basic derivation formula
(2) Four operations of derivatives
(3) Derivative of composite function
Let it be derivable at point x and y= derivable at point, then the composite function is derivable at point x, i.e.
Second, about the limit.
. 1. sequence limit:
Roughly speaking, when the number of n items in a series increases infinitely, the number of items in the series tends to infinity, which is the descriptive definition of the limit of the series. Write it down = a. For example:
The limit of 2 functions:
When the independent variable X approaches a constant infinitely, if the function approaches a constant infinitely, it is said that when X approaches, the limit of the function is written as
Third, the concept of derivative.
1, the derivative is in.
2. Derivative of.
3. The geometric meaning of the derivative of the function at this point:
The derivative of the function at this point is the slope of the tangent of the curve at this point,
That is, k=, and the corresponding tangent equation is
Note: the function value of the derivative function at is the derivative at.
For example, if =2, then = () a-1b-2c1d.
Fourth, the comprehensive application of derivatives
(a) the tangent of the curve
The derivative of the function y=f(x) at this point is the slope of the tangent of the curve y=(x) at this point. Therefore, the derivative can be used to solve the tangent equation of the curve. The specific solution is divided into two steps:
(1) Find the derivative of the function y=f(x) at this point, that is, the slope k =； of the tangent of the curve y=f(x) at this point;
(2) The tangent coordinates and the tangent slope are known, and the tangent equation is _.
Summary and sharing of knowledge points of mathematical functions and derivatives in senior high school;
First, the problem of finding the domain of a function ignores that the domain of a detailed function is the range of independent variables that make the function meaningful. If candidates want to find the definition domain accurately in the examination room, they should find out the restrictive conditions of independent variables in various situations according to the resolution function and list them into inequality groups. The solution set of inequality groups is the definition domain of the function. When finding the domain of a general function, we should pay attention to the following points: the denominator is not 0; Even times are nonnegative open; The real number is greater than 0, and the power of 0 is meaningless. The domain of a function is a set of non-empty numbers, so don't forget this when solving the problem of the domain of a function. For compound functions, it should be noted that the domain of external function is determined by the domain of internal function.
Second, it is wrong to judge the monotonicity of a function by its absolute value. Functions with absolute values are essentially piecewise functions. There are two ways to judge the monotonicity of piecewise function: one is to find the monotonous interval on each segment according to the monotonicity of the function represented by resolution function, and then integrate the monotonous interval on each segment; Second, draw the image of this piecewise function, and make an intuitive judgment by combining the image and nature of the function. Function problems cannot be separated from function images, which reflect all the properties of functions. When solving function problems, candidates should draw function images in their minds at the first time, analyze and solve problems from the images. For monotone increase (decrease) intervals of different functions, remember not to use union, just indicate that these intervals are monotone increase (decrease) intervals of functions.
Third, the common mistakes in finding function parity The most common mistakes in finding function parity are: finding the wrong function definition domain or ignoring the function definition domain, unclear preconditions for function parity, improper judgment methods for piecewise function parity, and so on. To judge the parity of a function, we must first consider the domain of the function. The necessary condition for a function to have parity is that the domain interval of the function is symmetric about the origin. If this condition is not met, the function must be an odd or even function. On the premise that the domain interval is symmetrical about the origin, the judgment is made according to the definition of parity function. When judging by definition, we should pay attention to the arbitrariness of independent variables in the definition domain.
Fourth, the reasoning of abstract functions is not rigorous. Many abstract function problems are designed by abstracting the * * * and "characteristics" of a certain function. When solving this kind of problems, candidates can solve abstract functions by analogy with the properties of some specific functions in this kind of functions. Using special assignment method, we can find the invariant property of function through special assignment, which is often the breakthrough of the problem. The proof of the properties of abstract functions belongs to algebraic reasoning. Like the proof of geometric reasoning, candidates should pay attention to the rigor of reasoning when answering questions. Every step should have sufficient conditions, don't leave out conditions, and don't make up conditions. The reasoning process is distinct, and attention should be paid to writing norms.
Fifthly, the function zero theorem is not applied properly. If the image of the function y=f(x) in the interval [a, b] is a continuous curve f (a) f (b)
Sixth, the tangent of a point on the two tangent curves refers to the tangent of the curve with this point as the tangent point, and there is only one such tangent; Tangents of a curve passing through a point refer to all tangents of the curve passing through that point. If this point on the curve certainly includes the tangent of the curve at this point, there may be more than one tangent of the curve passing through this point. Therefore, when solving the tangent problem of curve, candidates should first distinguish what is tangent.
Seventh, confuse the relationship between derivative and monotonicity. Function is a question type that increases function in a certain interval. If the examinee thinks that the derivative function of the function is always greater than 0 in this interval, it is easy to make mistakes. When solving the relationship between monotonicity of a function and its derivative function, we must pay attention to the necessary and sufficient condition that the derivative function of the function monotonically increases (decreases) in a certain interval is that the derivative function of the function is constant (small) or equal to 0 in this interval, and the derivative function is not constant in any subinterval of this interval.
Eighth, the relationship between derivative and extreme value is unclear. When solving the function extreme value problem with derivative, candidates are prone to make mistakes, that is, they find the points that make the derivative function equal to 0, but they don't judge the signs of the derivative functions on the left and right sides of these points. They mistakenly think that the point where the derivative function is equal to 0 is the extreme point of the function, and often make mistakes. The reason for the error is that the examinee is not clear about the relationship between derivative and extreme value. The zero derivative function value of a differentiable function at a certain point is only a necessary condition for the function to take the extreme value at that point. I would like to remind the majority of candidates that when calculating the extreme value of a function with derivatives, we must carefully check the extreme point.