2. The derivative formula of basic elementary function can be used to find the derivative of simple function.
Guide to learning methods 1. Derive the derivative formula of simple function by using the definition of derivative, and compare the derivative formula of general polynomial function to realize the idea from special to general. By clarifying the process of derivative, we can cultivate the ability of induction and exploration of laws and improve our interest in learning.
2. The formula in this section is the basis of the following lessons, and memorizing the formula correctly is the key to learning the content of this chapter well. When memorizing formulas, we should pay attention to the relationship between formulas. For example, Formula 6 is a special case of Formula 5, Formula 8 is a special case of Formula 7, and ln a has different positions in Formula 5 and Formula 7.
1. Derivatives of several common functions
Derivative function of original function
f(x)=c f？ (x)= 1
f(x)=x f？ (x)= 1
f(x)=x2 f？ (x)= 1
f(x)= 1x
f？ (x)= 1
f(x)=x
f？ (x)= 1
2. Derivative formula of basic elementary function
Derivative function of original function
f(x)=c f？ (x)= 1
f(x)=x？ (Q*) f？ (x)= 1
f(x)=sin x f？ (x)= 1
f(x)=cos x f？ (x)= 1
f(x)=ax f？ (x)=(a & gt； 0)
f(x)=ex f？ (x)= 1
f(x)=logax
f？ (x)=(a & gt； 0 and a? 1)
f(x)=ln x f？ (x)= 1
Explore the derivatives of several commonly used functions at the first point.
Question 1 How to find the derivative of the function y=f(x) by definition?
Question 2: Find the derivative of the following common functions by definition: (1) y = c (2) y = x (3) y = x2 (4) y =1x (5) y = x.
The geometric meaning of the derivative of question 3 is the slope of the tangent of the curve at a certain point. The physical meaning is the instantaneous speed of a moving object at a certain moment. What is the physical meaning of the derivative of a function (1) y =f(x)=c (constant)?
(2) What is the physical meaning of the derivative of the function y=f(x)=x?
Question 4: Draw an image of the function y =1x. According to the image, describe its change and find out the tangent equation of the curve at the point (1, 1).
Explore the derivative formula of the second basic elementary function
Question 1 The derivative function of a function can be obtained by using the definition of derivative, but the operation is complicated, and some function formulas cannot be deformed. How to solve this problem?
Question 2: Can you find the relationship between the derivative formulas of eight basic elementary functions?
Example 1 Find the derivative of the following function: (1)y=sin? 3; (2)y = 5x； (3)y = 1x 3； (4)y = 4x 3； (5)y =log3x。
Trace 1 and find the derivative of the following function: (1) y = x8; (2)y =( 12)x； (3)y = xx； (4)y=
Example 2 judges whether the following calculation is correct.
Find y=cos x in x=? Derive 3, the process is as follows: y? | = ? =-sin? 3=-32.
Trace 2 finds the derivative of the function f(x)= 13x at x= 1.
Comprehensive application of the third derivative formula of query point
Example 3 It is known that the straight line x-2y-4=0 intersects with the parabola y2=x at two points A and B, and O is the coordinate origin. Try to find a point P on the arc of parabola to maximize the area of △ABP.
Trace 3 points P is any point y=ex on the curve, and find the minimum distance from point P to straight line Y = X. 。
Standard test
1. Give the following conclusions: ① If y= 1x3, then y? =-3 x4； ② If y=3x, then y? = 133 x；
③ If y= 1x2, then y? =-2x-3； ④ If f(x)=3x, then f? (1)=3. The correct number is ().
A. 1
2. If the function f(x)=x, then f? (3) is equal to ()
A.36 B.0 C. 12x D.32
3. Let a point P on the sine curve y = y = sin x, and the tangent line with the point P as the tangent point is a straight line L, then the inclination range of the straight line L is ()
A.[0,? 4]? [3? 4,? )B.[0，？ )c .【？ 4,3? 4] D.[0，？ 4]? [? 2,3? 4]
4. The area of the triangle surrounded by the tangent of curve y=ex at point (2, e2) and the coordinate axis is _ _ _ _ _ _ _ _.