Derivative of Logarithm Function

 

Motivation:

 

                      f(x)=log(x),  where x > 0 .  Similarly, let’s start with the basic definition to find its derivative.

 

                      (log(x+h) – log(x))  =  log(  )

                                                           =  log( 1 +  ) =  log( 1 +  )

 

                     Up to this point, we find out it is difficult to go on.  Let’s see if we can transform it to other form:

                           log( 1 +  ) =

 

                   If  the limit of   exists when ,  then we can get a significant result. Otherwise, we need to find some other methods to find the derivative of f(x).  To find the limit of   exists as  ,  the problem is equivalent to the following one:

 

                               (1+  )ⁿ   as n  

 

Theorem ( Euler Number and Euler Series) :

                                The limit of  (1+  )ⁿ   exists as n  .

 

          Proof:

                         We use binomial theorem to represent it in the following way:

 

                         (1+  )ⁿ =  1+ n(  ) +  (  )²

                                           +  (  )³+ …+  (  )ⁿ

                                       1 + 1 +  +  + … +

 

                          And when we introduced Cauchy Criterion, we have seen that this series converges. It also can be proved by using ratio test. Thus,  (1+  )ⁿ  has a upper bound.

                   On the other hand,

 

                      (1+  )ⁿ =  1+ n(  ) +  (  )²

                                           +  (  )³+ …+  (  )ⁿ

                                    1 + 1 +  (  )² + … +  (  )^k

                          when  k < n.   The inequality holds by dropping some tailed positive terms.   Thus, for n goes to infinity,

 

                           (1+  )ⁿ  1+1+  +  + … +      for any k< n  .

 

             So,  (1+  )ⁿ  converges to  1 + 1 +  +  + … +   as n  .

             Usually, we denote this number as

 

                                                  e =  = 2.71828…

 

 

 

Derivative of Logarithm Function

 

                     Let’s continue the previous work.

 

                    (log(x+h) – log(x))  =

 

                    If we choose e as the base of the logarithm function,  and

 

                                             = e

                   then the result can be greatly simplified.  And we denote the logarithm function by using the base e as  “ln”;   e is known as the base of natural logarithm.

 

                   So,   for x>0,

                               

                                =

 

 

 

 

Theorem:    Let    f(x)=log_ax  ,  a  1,  x > 0 .  Then

                                 f’(x)=

 

              Proof:

 

                             f(x)= log_ax =

 

                           Thus,   f’(x) =  .

 

 

Example:   Find the derivative function of  f(x)=ln(2x²+1)

 

         Sol:

                        By chain rule,  we have

                                  f’(x) =  (4x) =