Gradient proof for implicitly defined function I am currently working on an assignment where I am seriously stuck. I need to prove that
$\nabla f(x) \cdot x = m \cdot f(x)$,
when $f: \mathbb{R^n} \rightarrow \mathbb{R}$ is differentiable and a $m \in \mathbb{N}$ with $f(t \cdot x) = t^m \cdot f(x)$ for all $t \neq 0$ and $x \in \mathbb{R^n}$ exists. So far I could deduce that $m=1$, since
$\nabla_x f(tx) = \nabla_x t^m f(x)$
$t \cdot \nabla_x f(x) = t^m \nabla_x f(x)$
$t = t^m$
$\Rightarrow m=1$
Furthermore I was able to show that f(0) = 0.
But now I am not sure how to proceed! Does anyone have an idea how I could go on?
Thanks for helping!

Hint:
In general for a differentiable $f:\Bbb{R}^n\to\Bbb{R}$, $x,v\in\Bbb{R}^n$, we have by the chain rule that
\begin{align}
\langle\nabla f(x), v\rangle = \dfrac{d}{ds}\bigg|_{s=0}f(x+sv).
\end{align}
Now, take $v=x$, and use homogeneity of $f$.
