Logarithmic derivative

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Calculus/MEmUTYoz">calculus</a> and <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, the logarithmic derivative of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> f is defined by the formula

f
              ′
            
            f
          
        
      
    
    {\displaystyle {\frac {f'}{f}}}

where 
 
 
 
 
 f
 ′
 
 
 
 {\displaystyle f'}
 
 is the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of f. Intuitively, this is the infinitesimal <a href="/facts/Relative_change/RrXmYBmn">relative change</a> in f; that is, the infinitesimal absolute change in f, namely 
 
 
 
 
 f
 ′
 
 ,
 
 
 {\displaystyle f',}
 
 scaled by the current value of f.
When f is a function f(x) of a real variable x, and takes <a href="/facts/Real_numbers/R02gw5Pb">real</a>, strictly <a href="/facts/Positive_number/z0xollg1">positive</a> values, this is equal to the derivative of ln(f), or the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> of f. This follows directly from the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a>:

d
            
              d
              x
            
          
        
        ln
        ⁡
        f
        (
        x
        )
        =
        
          
            1
            
              f
              (
              x
              )
            
          
        
        
          
            
              d
              f
              (
              x
              )
            
            
              d
              x
            
          
        
      
    
    {\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {1}{f(x)}}{\frac {df(x)}{dx}}}

Logarithmic derivative open-in-new

Logarithmic derivative