Homogenous function

A function $f$ is homogenous of degree $n$ if and only if $f(ax) = a^n f(x)$ .

— 1.8 Conserved Quantities

— Structure and Interpretation of Classical Mechanics

$\displaystyle{\begin{aligned} D_a f(ax) &= \frac{d}{da} f(ax) \\ &= \frac{d}{da} a^n f(x) \\ &= f(x) \frac{d}{da} a^n \\ &= n a^{n-1} f(x) \\ \end{aligned}}$

$\displaystyle{\begin{aligned} D_a f(ax) &= \frac{d}{da} f(ax) \\ &= \frac{d(ax)}{da} \frac{d}{d(ax)} f(ax) \\ &= x \frac{d}{d(ax)} f(ax) \\ &= x D_{ax} f(ax) \\ \end{aligned}}$