4.1 微分定義

 

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4.1 微分定義
4.2五大運算基本微分公式
4.3指數函數之微分
4.4對數函數之微分
4.5對數微分法

 

 

4-1 微分定義

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已知                            

割線斜率                      

切線斜率定義              

導數定義                      

 

 

將導數定義式變數變換        ,代入導數定義式

得導數之第二種形式         

再將式中取代得

導函數定義式              

其符號又可表成          

 

差分之定義                

導函數改成                  

微分之定義                

 

 

 

1.      Given a function ,. Find .

 

解答:

由導數定義   

            

        再將上式中取代得

       

 

2.      Given a function .  Find .

 

解答:

導函數定義式              

                                      

                                               

 

 

 

3.      Find the value of the derivative.  if

解答:

   

   

   

 

   

   

 

4.      Suppose  is differentiable at . Let  where  is a constant. Show that  is differentiable at  and evaluate  in terms of .

 

解答:

    ----------(1)

      ----------(2)

     Let , then

     (1) becomes

     (2) becomes

       is differentiable at    

     Hence

     Therefore,  is differentiable at  and .

 

 

5.   a. Let  be a function satisfying  for . Show that  is

     differentiable at  and find .

    b. Show that  is differentiable at  and find   .

 

解答:

     a.

        ----------(1)

        ----------(2)

         

         If  then

                   

                   

                   

           If  then

                   

                   

                   

       Therefore,  such that  is              

       differentiable at  and .

     b.

         when

         when

       Combining with , we can know  for all .

       By a., it can show that  is differentiable at  and .

 

 

6.      Does the parabola  have a tangent whose slope is  ? If so, find an equation for the line and the point of tangency. If not, why not ?

 

解答:

    

      

      

      

     We can solve  and .

     So, the point of tangency is  and the tangent equation is  

     .

 

 

7.   The figure shows the graph of a function over a closed interval . At what domain points does the function appear to be

     a. differentiable ?

     b. continuous but not differentiable ?                                                                               

     c. neither continuous nor differentiable ?

     Give reasons for your answers.

 

 

解答:

     a.

       At , we can not find the tangent of , so  is differentiable at 

        and .

     b.

       , so  is continuous but not differentiable.

     c.

       By a.,  is differentiable at  and , ie  is 

       continuous at  and .

       Combining with b., we can know  is continuous at .

       Therefore, we can not find such a domain in c.

 

 

 

 

 

 

 

 

首頁 | 4.1 微分定義 | 4.2五大運算基本微分公式 | 4.3指數函數之微分 | 4.4對數函數之微分 | 4.5對數微分法