4.4對數函數之微分

 

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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

 

  

4-4 對數函數之微分

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1. 對數函數之定義:    

 

對數特性                       

                                       

                                       

                                       

換底公式                       

2. 自然對數函數(Natural Logarithmic Function):

【定義一】

代入一般對數函數,定義

       

【定義二】

 

Definition of the Natural Logarithmic Function

       

 

對數特性(Logarithmic Property)

                                       

                                       

                                       

                                       

                                       

                                         

換底公式                        ,當

 

自然對數函數之圖形

                                      

 

3.自然對數函數之微分

 

自然對數函數之定義推導得       

                                  

                                  

 

4.一般對數函數之微分

 

【證明】

已知                              

代數上式對數微分得  

                                  

                                  

 

1.      求函數之導數?

 

解答:應用連鎖律

,則

        連鎖律

 

 

2.      方程式的解?

 

解答:

 已知             

依對數特性  

移項              

                       

 

3.    之導數?

 

解答:

由一般對數函數之微分        

 

由連鎖律     

化簡      

 

4.      Differentiate the function .

 

解答:

    

                    

                      .

 

 

5.      Find  for .

 

解答:

     Take the logarithm of both sides of the equation . We obtain

                   

                 

                 

    

           

          

            .

 

      

6.      Find the derivative of  given by .

 

解答:

    

       

         .

 

 

7.      Suppose the air resistance to an object dropped from height  is proportional to the square of its velocity (with proportionality constant ). Its height at time  is given by .

 

解答:

    

        

         .

 

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