Not all functions have derivatives, and a function does not necessarily have derivatives at all points. If the derivative of a function exists at a certain point, it is said to be derivative at this point, otherwise it is called non-derivative. However, the differentiable function must be continuous; Discontinuous functions must be non-differentiable.
For differentiable function f(x), x? F'(x) is also a function called the derivative function of f(x). The process of finding the derivative of a known function at a certain point or its derivative function is called derivative. Derivative is essentially a process of finding the limit, and the four algorithms of derivative also come from the four algorithms of limit.
Characteristics of derivative products:
(1) If the derivative is greater than zero, it will increase monotonically; If the derivative is less than zero, it decreases monotonically; The derivative equal to zero is the stagnation point of the function, not necessarily the extreme point. To judge monotonicity, the derivatives of the left and right values of the entry point are required.
(2) If the known function is increasing function, the derivative is greater than or equal to zero; If the known function is a subtraction function, the derivative is less than or equal to zero.
If the derivative function of a function is always greater than zero (or always less than zero) in a certain interval, then the function monotonically increases (or monotonically decreases) in this interval, which is also called the monotonic interval of the function.
The point where the derivative function is equal to zero is called the stagnation point of the function, and at this point, the function may get the maximum or minimum value (that is, the extreme value suspicious point). Further judgment needs to know the sign of the nearby derivative function. For a satisfaction point, if it exists such that it is greater than or equal to zero in the preceding interval and less than or equal to zero in the following interval, it is a maximum point, and vice versa.