1. What is a derivative?
The derivative is the "average rate of change" △y/△x ",which is the limit value when △x→0". The derivative value of differentiable function y=f(x) at point (a, b) is f'(a).
Second, the derivative formula of basic elementary function
The types of functions involved in the derivative formula of basic elementary functions in senior high school mathematics are: constant function, power function, sine function, cosine function, exponential function and logarithmic function. Their derivative formulas are shown in the following figure:
Derivative formula of basic elementary function in high school mathematics
Three, derivative addition, subtraction, multiplication and division.
The formulas of the four algorithms of derivative addition, subtraction, multiplication and division are shown in the following figure:
1, addition and subtraction algorithm
Derivative addition and subtraction algorithm formula
2. Multiplication and division algorithm
Derivative multiplication and division algorithm formula
Note the denominator g(x)≠0.
In order to facilitate memory, we can simplify the four algorithms of derivatives into four simple formulas as shown in the figure below.
Simplified formulas of four derivative algorithms
Note the denominator v≠0.
Fourth, the derivative formula of composite function ("chain rule")
To find the derivative of a basic elementary function, just substitute it into the "derivative formula of basic elementary function". For the derivation of functions other than basic elementary functions, such as "y=sin(2x)", the derivation rule of compound functions (also called "chain rule") should be used. Its contents are as follows.
(1) If the function y=f(g(x)), the relationship between its derivative and the derivatives of the functions y=f(u) and u=g(x) is shown in the following figure.
Derivative formula of compound function
(2) According to the derivative formula of the compound function, "the derivative of Y to X is equal to the product of the derivative of Y to U and the derivative of U to X".
Find the derivative of y=sin(2x).
Solution: y=sin(2x) can be regarded as a composite function of y=sinu and u=2x.
Because (sinu)'=cosu, (2x)'=2,
Therefore, [sin(2x)]'=(sinu)'×(2x)'
=cosu×2=2cosu=2cos(2x).
5. Physical meaning and geometric meaning of derivative value of differentiable function at a point.
(1) Physical meaning: instantaneous change rate of differentiable function at this point.
(2) Geometric meaning: the tangent slope value of differentiable function at this point.
Note that the derivatives of the linear function "kx+b(k≠0)" are all equal to the slope "k", that is, (kx+b)' = k.