In calculus, 
the racetrack principle describes the movement and growth of two functions in terms of their derivatives.
 This principle is derived from the fact that if a horse named Frank Fleetfeet always runs faster than a horse named Greg Gooseleg, then if Frank and Greg start a race from the same place and the same time, then Frank will win.  More briefly, the horse that starts fast and stays fast wins.
 In symbols: 
 or, substituting ≥ for > produces the theorem
 which can be proved in a similar way
 This principle can be proven by considering the function 
  
    
      
        h
        (
        x
        )
        =
        f
        (
        x
        )
        −
        g
        (
        x
        )
      
    
    {\displaystyle h(x)=f(x)-g(x)}
  
. If we were to take the derivative we would notice that for 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}
  
,
 Also notice that 
  
    
      
        h
        (
        0
        )
        =
        0
      
    
    {\displaystyle h(0)=0}
  
. Combining these observations, we can use the mean value theorem on the interval 
  
    
      
        [
        0
        ,
        x
        ]
      
    
    {\displaystyle [0,x]}
  
 and get
 By assumption, 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}
  
, so multiplying both sides by 
  
    
      
        x
      
    
    {\displaystyle x}
  
 gives 
  
    
      
        f
        (
        x
        )
        −
        g
        (
        x
        )
        >
        0
      
    
    {\displaystyle f(x)-g(x)>0}
  
. This implies 
  
    
      
        f
        (
        x
        )
        >
        g
        (
        x
        )
      
    
    {\displaystyle f(x)>g(x)}
  
.
 The statement of the racetrack principle can slightly generalized as follows; 
 as above, substituting ≥ for > produces the theorem 
 This generalization can be proved from the racetrack principle as follows:
 Consider functions 
  
    
      
        
          f
          
            2
          
        
        (
        x
        )
        =
        f
        (
        x
        +
        a
        )
      
    
    {\displaystyle f_{2}(x)=f(x+a)}
  
 and 
  
    
      
        
          g
          
            2
          
        
        (
        x
        )
        =
        g
        (
        x
        +
        a
        )
      
    
    {\displaystyle g_{2}(x)=g(x+a)}
  
.
Given that 
  
    
      
        
          f
          ′
        
        (
        x
        )
        >
        
          g
          ′
        
        (
        x
        )
      
    
    {\displaystyle f'(x)>g'(x)}
  
 for all 
  
    
      
        x
        >
        a
      
    
    {\displaystyle x>a}
  
, and 
  
    
      
        f
        (
        a
        )
        =
        g
        (
        a
        )
      
    
    {\displaystyle f(a)=g(a)}
  
,
 
  
    
      
        
          f
          
            2
          
          ′
        
        (
        x
        )
        >
        
          g
          
            2
          
          ′
        
        (
        x
        )
      
    
    {\displaystyle f_{2}'(x)>g_{2}'(x)}
  
 for all 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}
  
, and 
  
    
      
        
          f
          
            2
          
        
        (
        0
        )
        =
        
          g
          
            2
          
        
        (
        0
        )
      
    
    {\displaystyle f_{2}(0)=g_{2}(0)}
  
, which by the proof of the racetrack principle above means 
  
    
      
        
          f
          
            2
          
        
        (
        x
        )
        >
        
          g
          
            2
          
        
        (
        x
        )
      
    
    {\displaystyle f_{2}(x)>g_{2}(x)}
  
 for all 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}
  
 so 
  
    
      
        f
        (
        x
        )
        >
        g
        (
        x
        )
      
    
    {\displaystyle f(x)>g(x)}
  
 for all 
  
    
      
        x
        >
        a
      
    
    {\displaystyle x>a}
  
.
 The racetrack principle can be used to prove a lemma necessary to show that the exponential function grows faster than any power function. The lemma required is that
 for all real 
  
    
      
        x
      
    
    {\displaystyle x}
  
. This is obvious for 
  
    
      
        x
        <
        0
      
    
    {\displaystyle x<0}
  
 but the racetrack principle is required for 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}
  
. To see how it is used we consider the functions
 and
 Notice that 
  
    
      
        f
        (
        0
        )
        =
        g
        (
        0
        )
      
    
    {\displaystyle f(0)=g(0)}
  
 and that
 because the exponential function is always increasing (monotonic) so 
  
    
      
        
          f
          ′
        
        (
        x
        )
        >
        
          g
          ′
        
        (
        x
        )
      
    
    {\displaystyle f'(x)>g'(x)}
  
. Thus by the racetrack principle 
  
    
      
        f
        (
        x
        )
        >
        g
        (
        x
        )
      
    
    {\displaystyle f(x)>g(x)}
  
. Thus,
 for all 
  
    
      
        x
        >
        0
      
    
    {\displaystyle x>0}
  
.