4.3指數函數之微分

 

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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

 

  

4-3 指數函數之微分 

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1.指數函數(Exponential Function)之定義:

  

   為以為底之指數函數。

指數特性                       

                                       

                                       

                                       

 

2.

同理

指數特性                       

                                       

                                       

                                       

3.自然指數函數的導數

  的可微函數

 

 

 

4. 一般指數函數之微分

已知指數函數               

代換成自然指數函數   

微分(連鎖律)           

 

微分公式                     

 

連鎖律                           

 

1.      Differentiate 

解答:

 ,則

 

2.      Differentiate 

解答:

除法法則                  

     

    化簡         

 

3.      Let  for every positive integer . Find  in terms of  and .

 

解答:

    

          .

 

 

4.      Calculate the derivative of .

 

解答:

    

         

          .

 

 

5.      Calculate the derivative of .

 

解答:

    

         

         .

 

 

6.      Suppose  and  are positive constants. If a variable  grows exponentially according to , then the doubling time  is defined by the property that the value of  doubles when  is increased by . Express the rate constant  in terms of the doubling time  of .

 

解答:

    

     The rate constant of change of  at time  is

                      .

      

     Therefore, we can solve .

 

 

 

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