Knowledge points of mathematical derivatives in senior two.
1. Find the monotonicity of a function:
The basic method of finding monotonicity of function by using derivative: Let function yf(x) be derivable in the interval (a, b), (1) If f(x) is constant, then function yf(x) is increasing function in the interval (a, b); (2) If F (x) is a constant, the function yf(x) is a decreasing function in the interval (a, b); (3) If f (x) is constant, then the function yf(x) is a constant function in the interval (a, b).
The basic steps of finding monotonicity of function by derivative are as follows: ① finding the domain of function yf(x); ② Find the derivative of f(x); ③ Solve the inequality f(x)0, and define the uninterrupted interval of the solution set on the domain as the increasing interval; ④ If the inequality f(x)0 is solved, the uninterrupted interval of the solution set on the domain is a decreasing interval.
Conversely, we can also use derivatives to solve related problems (such as determining the range of parameters) through the monotonicity of functions: let function yf(x) be derivable in the interval (a, b),
(1) If the function yf(x) is increasing function in the interval (a, b), then f(x)0 (where the x value of f(x)0 does not constitute the interval);
(2) If the function yf(x) is a subtraction function in the interval (a, b), then f(x)0 (where the x value of f(x)0 does not constitute the interval);
(3) If the function yf(x) is a constant function in the interval (a, b), then f(x)0 holds. 2. Find the extreme value of the function:
Let the function yf(x) be defined in x0 and its vicinity. If all points near x0 have f(x)f(x0) (or f(x)f(x0)), it is said that f(x0) is the minimum (or maximum) of the function f(x).
The extreme value of differentiable function can be obtained by studying the monotonicity of function. The basic steps are as follows:
(1) Determine the domain of the function f(x); (2) find the derivative f (x); (3) Find all the real roots of the equation f(x)0, x 1x2xn, divide the domain into several cells in sequence, and list the values of f(x) and f(x) when x changes.
Change:
(4) Look up the sign of f(x) and judge the extreme value from the table. 3. Find the maximum and minimum values of the function:
If the function f(x) has x0 in the domain I, so that there is always f(x)f(x0) for any xI, it is said that f(x0) is the maximum value of the function in the domain. The extreme value of a function in the definition domain is not necessarily unique, but the maximum value in the definition domain is unique.
Find the maximum and minimum value of the function f(x) in the interval.
Comments: This topic examines the basic knowledge of parity, monotonicity, the maximum value of quadratic function, the application of derivative, and the ability of reasoning and operation.
Mathematical derivative formula of senior two
1.①
②
③
2. Derivative relationship between the original function and the inverse function (the inverse trigonometric function is derived from the derivative of trigonometric function): If the inverse function of y=f(x) is x=g(y), then there is y'= 1/x'.
3. Derivative of composite function:
The derivative of the compound function to the independent variable is equal to the derivative of the known function to the intermediate variable, multiplied by the derivative of the intermediate variable to the independent variable-called the chain rule.
4. The integral derivative rule:
(a(x), b(x) is a subfunction)
Calculation of derivative
The derivative function of a known function can be calculated by using the limit of change rate according to the definition of derivative. In practical calculation, most common analytic functions can be regarded as the result of sum, difference, product, quotient or mutual compound of some simple functions. As long as the derivative functions of these simple functions are known, the derivative functions of more complex functions can be calculated according to the derivative law.
Derivation rule of derivative
Deduction rule
The derivative function of a function composed of the sum, difference, product, quotient or mutual combination of basic functions can be derived from the derivative rule of the function. The basic deduction rules are as follows:
Linearity of derivative: To find the linear combination of derivative functions is equivalent to finding the derivatives of each part first and then finding the linear combination.
The derivative function of the product of two functions, one derivative multiplied by two+one derivative multiplied by two.
The derivative function of the quotient of two functions is also a fraction. (derivative times mother-derivative times mother) divided by mother.
Derivation rule of compound function
If there is a compound function, if you need the derivative of the function at a certain point, you can first find the derivative function of this function by the above method, and then you can see the value of the derivative function at this point.
higher derivative
Solution of higher derivative
1. Direct method: Find the higher derivative step by step from its definition.
Generally used to find solutions to problems.
2. The algorithm of higher derivative:
(binomial theorem)
3. Indirect method: using the known higher-order derivative formula, through four operations, variable substitution and other methods.
Note: the function after substitution should be easy to find, and try to get the order derivative as close as possible to the known formula.