I want to prove that if $f(x,y)$ has continuous partial derivatives then: $$\forall x,y,t>0\ f(x,y)=f(tx,ty)\iff x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=0\ \text{for every}\ x,y>0$$
In this $\rightarrow$ direction I tried using chain rule but didn't see exactly how it helps me. Any help would be appreciated.
Fix $(x,y) \in \mathbb{R}^2$ and consider the curve $$\gamma : (0,\infty) \rightarrow \mathbb{R}^2$$ $$ t \mapsto (tx,ty)$$ And observe that $(f \circ \gamma)’(t) = 0$ for every $t \in (0,\infty)$, then the map $t \mapsto f(tx,ty)$ is constant for every $(x,y) \in \mathbb{R}^2 $