Derivative is an important basic concept in calculus. The limit of the quotient between the increment of the dependent variable and the increment of the independent variable when the increment of the independent variable tends to zero. When a function has a derivative, it is said to be derivative or differentiable. The differentiable function must be continuous. Discontinuous functions must be non-differentiable. Derivative is essentially a process of finding the limit, and the four algorithms of derivative come from the four algorithms of limit. Also known as Ji Shu and WeChat services, they are mathematical concepts abstracted from the problems of speed change and curve tangent. Also known as the rate of change. For example, if a car walks 600 kilometers per hour/kloc-0, its average speed is 60 kilometers per hour, but in the actual driving process, the speed changes, not all of them are 60 kilometers per hour. In order to better reflect the speed change of the car during driving, the time interval can be shortened. Let the relationship between the position S of the car and time t be S = f (t), then the average speed of the car during the period from time t0 to t 1 is [f (t1)-f (t0)]/[t1-t0]. The speed change of the car will not be great, and the average speed can better reflect the movement change of the car from t0 to t 1. Naturally, the limit [f (t1)-f (t0)]/[t1-t0] is taken as the instantaneous speed of the car at t0, which is also commonly known as the speed. Generally speaking, if the unary function y = f (x) is defined near the point x0 (x0-a, x0+a), when the increment of the independent variable δ x = x-x0 → 0, the limit of the ratio of the function increment δ y = f (x)-f (x0) to the increment of the independent variable exists and is limited, so the function f is derivable at the point x0. If the function f is differentiable at every point in the interval I, a new function with I as the domain is obtained, which is called f', called the derivative function of f, or derivative for short. The geometric meaning of the derivative f'(x0) of the function y = f (x) at x0: it represents the tangent slope of the curve l at P0 [x0, f (x0)]. Generally speaking, we get the law of judging the increase or decrease of function by derivative: let y = f (x) be derivable in (a, b). If in (a, b) and f' (x) >: 0, then f(x) increases monotonically in this interval. . If in (a, b), f' (x) 0 and a is not equal to/kloc-. The above formula can not replace constants, but only functions. People who are new to derivatives often ignore this point, which leads to ambiguity. We should pay more attention to it. (3) Four algorithms of derivative: ① (Uv)' = Uv' ② (UV)' = Uv+UV' ③ (U/V)' = (Uv-UV')/V2 (4) The derivative of the compound function to the independent variable is equal to the known function. Derivative is an important pillar of calculus. Newton and Leibniz made outstanding contributions to this! Application of derivative 1. Monotonicity of function (1) uses the sign of derivative to judge the increase or decrease of function, and uses the sign of derivative to judge the increase or decrease of function. This is an application of the geometric meaning of derivative in studying the law of curve change, which fully embodies the idea of combining numbers with shapes. Generally in a certain interval (a, b), if > 0, then the function y = f (. If < 0, the function y=f(x) monotonically decreases in this interval. If there is always =0 in an interval, then f(x) is a constant function. Note: In a certain interval, > 0 is a sufficient condition for f(x) to be increasing function in this interval, but it is not a necessary condition. For example, f(x)=x3 is increasing function, but .. ② is derived; (3) Use (or) to solve the range corresponding to x. When f' (x) > 0, f(x) is increasing function in the corresponding interval; When f' (x) < 0, f(x) is a decreasing function in the corresponding interval. 2. Determination of extreme value of function (1) ① If both sides have the same sign, it is not the extreme point of f(x); (2) If it is on the left or right side nearby, then it is the maximum or minimum. (3) The step of finding the extreme value of the function (1) determines the definition domain of the function; ② Deduction; ③ Find all the stationary points in the definition domain, that is, find all the real roots of the equation; (4) Check the symbols around the stagnation point. If Zuo Zheng is negative to the right, then f(x) gets the maximum at this root; If the left side is negative and the right side is positive, then f(x) gets the minimum at this root. 4. Maximum value of function (1) If the maximum value (or minimum value) of f(x) on [a, b] is obtained at one point in (a, b), obviously this maximum value (or minimum value) is also a maximum value (or minimum value). But the maximum value can also be obtained at the end point A or B of [a, b], and extreme value and maximum value are two different concepts. (2) the step of finding the maximum and minimum value of f(x) in [a, b] 1 finding the extreme value of f(x) in [a, b]; ② Compare the extreme values of f(x) with those of f(a) and f(b), and the maximum value is the maximum value, and the minimum value is the minimum value. 5. In life, we often encounter optimization problems such as profit maximization, material saving and highest efficiency. These problems are called optimization problems, and optimization problems are also called maximum problems. Solving these problems has important practical significance. These problems can usually be translated into.