Racetrack principle
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 Proof. This principle can be proven by considering  the function h(x) = f(x) - g(x). If we were to take the derivative we would notice that for x>0 Also notice that h(0) = 0. Combining these observations, we can use the mean value theorem on the interval [0, x] and get By assumption, formula_2, so multiplying both sides by formula_14 gives f(x) - g(x) > 0. This implies f(x) > g(x). Generalizations. The statement of the racetrack principle can slightly generalized as follows; as above, substituting ≥ for > produces the theorem Proof. This generalization can be proved from the racetrack principle as follows: Consider functions formula_25 and formula_26. Given  that formula_1 for all formula_16, and formula_17, formula_30 for all formula_2, and formula_32, which by the proof of the racetrack principle above means formula_33 for all formula_2 so formula_4 for all formula_16. Application. 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. This is obvious for x<0 but the racetrack principle is required for x>0. To see how it is used we consider the functions and Notice that f(0) = g(0) and that because the exponential function  is always increasing (monotonic) so formula_1. Thus by the racetrack principle f(x)>g(x). Thus, for all x>0.