Summary of knowledge points

Average change rate of function, instantaneous change rate of function, concept of derivative, general steps of derivative function, geometric meaning of derivative, definition and calculation of derivative, derivative addition (subtraction) rule, derivative multiplication rule, derivative division rule, derivative of simple compound function and other knowledge points. Among them, understanding the definition of derivative is the key, and at the same time, we should memorize the derivatives of eight commonly used functions and their algorithms.

Common inspection methods

The stage exam examines the knowledge of derivatives in the form of multiple-choice questions, fill-in-the-blank questions and solution questions. In the college entrance examination, the knowledge of derivatives is mainly combined with the problems solved by functions. General deduction is easy to answer. Solve the problem directly by derivative algorithm and derivative method of compound function.

(A) the first definition of derivative

Let the function y=f(x) be defined in a domain of point x0. When the independent variable x has increment △ x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets increment △ y = f (x0+△ x)-f (x0); If the ratio of △y to △x has a limit when △x→0, the function y=f(x) can be derived at point x0, and this limit value is called the derivative of function y=f(x) at point x0, which is also called f'(x0), which is the first definition of derivative.

(2) The second definition of derivative

Let the function y=f(x) be defined in a domain of point x0. When the independent variable x changes △ x at x0 (x-x0 is also in the neighborhood), the function changes △y=f(x)-f(x0) accordingly. If the ratio of △y to △x is limited when △x→0, then the function y=f(x) is derivable at point x0. This limit value is called that the derivative of function y=f(x) at point x0 is f'(x0), which is the second definition of derivative.

(3) Derivative function and derivative

If the function y=f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y=f(x) corresponds to a certain derivative of each certain value of x in the interval I, and forms a new function, which is called the derivative function of the original function y=f(x), and is denoted as y' and f'. Derivative function is called derivative for short.

Monotonicity and its application

1. General steps of studying monotonicity of polynomial function with derivative

(1) Find f? (10)

(2) determine f? (x) If the symbol (3) in (a, b) is f? (x)>0 is a constant on (a, b), then f(x) is a increasing function on (a, b); If f? (x) The corresponding interval of the intersection of solution set 0 and the domain is an increasing interval; f？ (10)