I. Introduction:
Used to describe the properties of the derivative of a function in an interval. This theorem shows that if a function is differentiable in an interval, its derivative will take all the values between the values of the derivative at the end of the interval.
Specifically, let the function f(x) be continuous in the closed interval [a, b] and differentiable in the open interval (a, b). Then for any c between f'(a) and f'(b), there is a point x0 in the open interval (a, b) that makes f' (x0) = c.
Second, the derivative:
Derivative is an important concept in calculus, which is used to describe the rate of change or slope of a function at a certain point. It is the instantaneous rate of change of a function at each point, which is usually expressed as the derivative of the function f(x) to the independent variable x, and is recorded as f'(x) or dy/dx.
If the derivative of the function f(x) exists at a certain point x0, then the derivative can be expressed by the following limit definition:
[f '(x _ 0)= \ lim _ { h \ to 0 } \ frac { f(x _ 0+h)-f(x _ 0)} { h }]
Where x0 is a point and h is a real number close to zero.
The concept and properties of derivative;
First, the geometric meaning of derivative:
The derivative of a function at a certain point is equal to the slope of the tangent of the curve at that point, which describes the instantaneous rate of change of the function at that point.
Second, the sign of the derivative:
If the derivative is positive, it means that the function is increasing at this point; If the derivative is negative, the function decreases at this point; If the derivative is zero, it means that the function obtains local extremum at this point.
Third, the calculation rules of derivative:
There are a series of derivative calculation rules, such as constant rule, power rule, sum and difference rule, product rule, quotient rule, etc., which are used to calculate the derivative of complex functions.
Fourth, the higher derivative:
Besides the first derivative, we can also define the second derivative, the third derivative and so on. , which represents the derivative of the derivative of the function and describes the acceleration of the function.
Five, the application of derivative:
Derivatives are widely used in many fields, such as physics, engineering, economy and so on. It can be used to solve optimization problems, draw curves, calculate speed and acceleration, etc.