If f satisfies (40) for all x, it is homogeneous of degree k.
Euler's theorem can be thought of as a multidimensional extension of the rule for the derivative of a power function (example 4.15). Two special cases are worth considering. When n = 1, equation (40) becomes

which is precisely the rule for a power function (example 4.15). When k = 1, equation (40) becomes

In this case the derivative is exact (rather than an approximation) along any ray through the origin. We give a sample of applications of Euler's theorem in economics.

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kf(x) = f' (x)x-f"(x) = km Drix)(x) Σο,f(x]Xi-/(x) for every x e S