Differentiate ln x

                    
Since   


Then the derivative of ln x is 1/x,  x > 0   (Fundamental Theorem)

Derivative of ln u, where u = f(x):

• If y = ln |u|,  then   y' = u'
                                       u
  
  Proof:
• If  u > 0, then ln |u| = ln u  and   y' = u'
                                                             u
• If  u < 0, then ln |u| = ln (-u)  and   y' = -u'   =   u'
                                                                -u          u
  
  

Examples:

1. y = ln 3x
   y' = 3(1/3x) = 1/x
2. y =    (x + 2)²
          \/(x² - 1)
1. Find the natural logarithm of both sides:
   ln y = ln    (x + 2)²
                   \/(x² - 1)
2. Simplify, using laws of logarithms:
   ln y = 2 ln (x + 2) - 1/2 ln (x² - 1)
3. Differentiate both sides:
   y'  =  2    1       -  1      2x     
               x + 2       2   (x² - 1)
4. Simplify:
   y'  =     2         -             x        
            x + 2               x² - 1
      
        =  2x² - 2 - x² - 2x
             (x + 2) (x² - 1)
   
        =     x² - 2x - 2     
            (x + 2) (x² - 1)

Problems