Theorem 1   Suppose is a differentiable positively homogeneous function of degree $r$ . Then $\frac{\partial f}{\partial x^i}$ is a positively homogeneous function of degree $r-1$ .

Proof. By considering component functions if necessary, we can assume that $m=1$ . For , let $M_\lambda$ be the multiplication map,

For $\lambda>0$ and , we have \begin{eqnarray*} \frac{\partial f}{\partial x^i}(\lambda v) &=&\frac{\partial(f\circ M_\lambda \circ M_{1/\lambda})}{\partial x^i}(\lambda v) \\ &= &\sum_{l=1}^n\frac{\partial(f\circ M_\lambda)}{\partial x^l} (v)\, \frac{ \partial(x\mapsto x/\lambda)^l}{\partial x^i} (\lambda v) \\ &= &\frac{\partial(f\circ M_\lambda)}{\partial x^i} (v)\, \frac{1}{\lambda}\\ &= &\lambda^{r-1}\frac{\partial f}{\partial x^i} (v) \end{eqnarray*}as claimed.