导数的概念及运算（Theconceptandoperationofderivative）

导数的概念及运算（The concept and operation of derivative）The concept and operation of derivative
The draft Code: Zhou reviewer: Yan Chunmei: Shanda publicity commissioning editor
Target cognition
Learning goals:
1. understand some actual background of derivative concept (such as instantaneous velocity, acceleration, the slope of tangent of smooth curve, etc.); grasp the function in one
The definition of derivative at point and the geometric meaning of derivative; understanding the concept of derivative.
2. memorize constant function y=C, power function y=xn (n is rational number), trigonometric function y=sinx, y=cosx, exponential function y=ex, y=ax, pair
The derivative formula of the number function y=lnx and y=logax; the four operation rule of derivative;
3. grasp the derivation rule of composite function, and find the derivatives of some simple composite functions.
A key：
The concept of derivative and its geometric meaning, the derivative formula of common function, the four operation rule
of derivative and the derivative formula of compound function.
Difficult point:
The concept of derivative and the derivative of compound function.
Learning strategy:
1. combining the concepts of average velocity, instantaneous velocity and acceleration in physics, the concept of derivative and the geometric meaning of derivative are understood.
2. remember the derivation formulas of elementary elementary functions, and the derivation rules of sum, difference, product and quotient, understand and grasp the derivation rule and formula knot
In order to solve the problem of derivative structure, the derivative operation can be carried out accurately and effectively.
3. pay attention to understand the derivation rule of composite function; for a composite function, we must clarify the compound relationship in the middle and clarify the decomposition function
Which variables should be dealt with in the derivation?.
Carding the main points of knowledge
Knowledge point one: the average change rate of function
(1) concept:
In the function, if there is an increase in the independent variable, then the function value y corresponding incremental Delta y=f (x0+ x) -f (x0), the ratio of the average change from x + delta function called rate, i.e..
If, then, the average rate of change can be expressed as the average change as a function from the call rate.
Be careful：
The rate of change is the ratio of the increments of the two quantities". The average expansion rate of a balloon is the increment of the radius and the volume increment
Ratio;
The average change rate of function shows the changing trend of function. When the value is smaller, the change of function can be accurately reflected.
(3) the amount of change of the independent variable at the location, but the change of the function value, which can be 0. Mean change rate of function
0, which does not necessarily mean that the function does not change, should be taken into account.
(2) geometric meaning of average change rate
The geometric meaning of the mean change rate of a function is the slope of the two-point secant on the connected function image.
As shown in the graph, the geometric meaning of the average change rate of the function is the slope of the line AB.
In fact，。

Function: according to the geometric meaning of the average change rate, the slope of the secant of the curve can be solved.
Knowledge point two: the concept of derivative:
The definition of 1. derivative:
For function, at the point to the independent variable x to increment, function y corresponding increments. If the limit exists, then the limit is called the derivative at the point, which is denoted as, or at this point, also derivable at the point.
Namely: (or)
Be careful：
Increments can be positive or negative;
The essence of derivative is the limit of the mean change rate
of a function at a certain point, that is, instantaneous change rate.
2. derivative function:
If the function has a derivative in every point in the interval, this time for each one, corresponds to a certain derivative, so as to form a new function, call this function function function in the open interval, referred to as the derivative.
Note: the derivative of a function and the derivative at the point are not the same concept. They are constants. They are the function values at the function and reflect the change of the function in the neighborhood.
Geometric meaning of 3. derivatives:
(1) tangent of curve
A point on the curve P (x0, Y0) and a point near Q (x0+ x, y0+ y, P, Q) after PQ for secant curve, the tilt angle is when the Q (x0+ x, y0+ y) along the curve closer to P (x0, Y0) that is, x, 0, PT PQ linear limit position called secant tangent curves at point P.
If the tangent of the tilt angle, when the delta x, 0, PQ is the limit of secant slope, the slope of the tangent line.
Namely:.
(2) the geometric meaning of derivative:
The derivative of a function at point x0 is the slope of the tangent at the point () on the curve.
Be careful：
If the derivative of the curve does not exist at the point, but there is a tangent, then the tangent is perpendicular to the axis.
The tangent and the positive axis angle, axial and tangent angle; the forward angle is an obtuse angle;,
Tangent parallel to axis.
(3) tangent equation of curve
If the point is derivable, the tangent equation of the curve at the point () is:
.
4. instantaneous velocity:
The speed of movement is equal to the displacement and time ratio, and non uniform linear motion in this ratio is changing, how to understand the movement speed of each moment of the non uniform linear motion, we adopt the concept of instantaneous velocity.
If the motion objects meet s=s (T) (displacement formula), then
the object at time t is the instantaneous velocity V, t objects to t+ t this time limit when the delta T 0, average speed, i.e..
If the function is regarded as the displacement formula of the object, the derivative represents the instantaneous velocity of the moving object at the moment.
Knowledge point three: the derivative formula of common basic functions
(1) (C is constant),
(2) (n is rational),
(3),
(4),
(5),
(6),
(7),
(8),
Knowledge point four: derivation rule of function four operations
Let all be conductive
(1) derivative of sum and difference:
(2) derivatives of product:
(3) the derivative of quotient: ()
Knowledge point five: the derivation rule of composite function
or
That is, the derivative of a composite function to an independent variable is equal to the derivative of the known function to the intermediate variable, and is multiplied by the derivative of the intermediate variable to the independent variable.
Attention: selection of intermediate variables is the key to the derivation of complex functions. It is necessary to remember the intermediate variable and divide the layer by layer in derivation. After derivation, the intermediate variable is converted into the function of the independent variable.
Law and method guidance
1. how to find the average change rate of function
The common "two steps" method is used to find the average change rate of function:
(1) the difference between the sum of the sum and the sum of
the sum
(2) quotient: the quotient of the obtained difference.
Be careful：
(1), the formula, the value may be positive or negative, but the value is not
For zero, the value can be zero. If the function is a constant function,.
(2) in the formula, and the corresponding "incremental", that is when,
.
(3) in the formula, when the fixed value, take a different value, the average change rate of the function is not
In the same way, the average change rate of the function is different when the fixed value is taken and different values are taken.
2. how to find the derivative of a function at one point
(1) the derivative is used to define the derivative of a function at one point, usually using the three step method".
Calculate the increment of function:;
Seeking the average change rate:;
Take the limit derivative:.
(2) the derivative of elementary function is obtained by using the derivative formula of basic elementary function.
The geometric meaning of 3. derivative
The derivative of the function at the point is the cut of the curve at the point ()
Line slope.
When the displacement is a function of time, it represents the instantaneous velocity of the object at the moment;
Third, the velocity is the function of time, and the acceleration of the object at the moment;
4. use the geometric meaning of derivative to find the tangent equation of the curve
Find the derivative at the place;
The tangent equation is obtained by using the point oblique equation of the linear equation.
5. general steps of finding derivatives of composite functions
Properly select intermediate variables and correctly decompose
complex relations;
Second, step by step derivation (to clarify which variables each step is derivative to which variables);
Third, the intermediate variable is replaced by the function of the original independent variable (generally x).
The whole process can be referred to as the decomposition of derivative to generation, skilled, you can omit the process. In case of multiple, can be repeatedly used intermediate variables.。