Prove the differential of $f$ at $x_1$

Question

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is homogenic from degree $p$ for every $t>0$ and $f$ is $C^{1}$ $$f(tx)=t^{p}f(x)$$ Prove $\frac{df}{dx_1}$ is homogenic from degree p-1

I've tried to use the chain rule with $\gamma(x)=x$

Answer 1

$t\frac {\partial } {\partial x_1} f(tx)=t^{p} \frac {\partial } {\partial x_1} f(x)$ by differentiation w.r.t. $x_1$. Just divide by $t$.