The Derivative of the Natural Logarithm

Derivation of the Derivative

Our next task is to determine what is the derivative of the natural logarithm.  We begin with the inverse definition.  If

        y  =  ln x

then

        e^y  =  x

Now implicitly take the derivative of both sides with respect to x remembering to multiply by dy/dx on the left hand side since it is given in terms of y not x.

        e^y dy/dx  =  1

From the inverse definition, we can substitute x in for e^y to get

       x dy/dx  =  1

Finally, divide by x to get

        dy/dx  =  1/x

We have proven the following theorem

Examples

Find the derivative of

        f(x)  =  ln(3x - 4)

Solution

We use the chain rule.  We have

        (3x - 4)'  =  3

and 

        (ln u)'  =  1/u  

Putting this together gives

        f '(x)  =  (3)(1/u)

                  3
         =                         
               3x - 4      

Example

find the derivative of 

        f(x)  =  ln[(1 + x)(1 + x²)²(1 + x³)³ ]

Solution

The last thing that we want to do is to use the product rule and chain rule multiple times.  Instead, we first simplify with properties of the natural logarithm.  We have

        ln[(1 + x)(1 + x²)²(1 + x³)³ ]  =  ln(1 + x) + ln(1 + x²)² + ln(1 + x³)³ 

        =  ln(1 + x) + 2 ln(1 + x²) + 3 ln(1 + x³)

Now the derivative is not so daunting.  We have use the chain rule to get

                            1                 4x                   9x²  
        f '(x)  =                   +                     +                      
                         1 + x            1 + x²             1 + x³

Exponentials and With Other Bases

              

Definition   Let a > 0 then           a x = ex ln a

Examples

Find the derivative of

        f (x) = 2^x

Solution

We write

        2^x = e^{x ln 2}

Now use the chain rule

        f '(x)  =  (e^{x ln 2})(ln 2)  =  2^x ln 2

Logs With Other Bases

    We define logarithms with other bases by the change of base formula.

         

Definition                          ln x        loga x =                                    ln a

    
Remark:  The nice part of this formula is that the denominator is a constant.  We do not have to use the quotient rule to find a derivative

Examples

Find the derivative of the following functions

1. f(x) = log₄ x 
   

2. f(x) = log (3x + 4)
   

3. f(x) = x log(2x) 
   

Solution

1. We use the formula
   
                        ln x    
           f(x)  =            
                        ln 4
   
   so that 
                             1    
           f '(x)  =              
                          x ln 4
   
   
   

2. We again use the formula
   
                          ln(3x + 4)
            f(x)  =                       
                            ln 10
   
          
   
   now use the chain rule to get
   
                                    3
            f '(x)  =                          
                           (3x + 4) ln 10
   
   

3. Use the product rule to get
   
           f '(x) = log(2x) + x(log(2x))'
   
   Now use the formula to get
   
                                ln(2x)
           log (2x) =                        
                                ln 10
   
   The chain rule gives
   
                                                   2                                   1
           f '(x)  =  log(2x) + x                       =  log(2x) +            
                                            (2x) ln 10                           ln 10
          
   

    

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