Let $m$ be a positive integer. Assume $f: R^n \to R$ has continuous first partials and satisfies the following property:
$f(t\vec{x})=t^mf(\vec{x})$ for all $\vec{x}$ and $t$
Show that $\vec{x} \cdot \nabla{f(\vec{x})}=mf(\vec{x})$
So far I have
$\vec{x} \cdot \nabla{f(\vec{x})}= \frac{1}{t^m}\vec{x} \cdot \nabla{f(t\vec{x})}$ but I'm not sure how to proceed from here.