Prove $\Bigg(\langle\nabla f(x),x \rangle = af(x) \Bigg) \Leftrightarrow \Bigg(f(tx)=t^af(x) \Bigg)$ for $f: \Bbb R^n \to \Bbb R$ differentiable

Question

Let $n\in \Bbb N , a \in \Bbb R$ and $f: \Bbb R^n \to \Bbb R$ differentiable. I have to show: $$\Bigg(\langle\nabla f(x),x \rangle = af(x) \:\:\:\forall x \in \Bbb R ^n \backslash\{ 0 \} \Bigg) \Leftrightarrow \Bigg(f(tx)=t^af(x) \:\:\:\forall t\gt 0 \textrm{ and }x \in \Bbb R ^n \backslash\{ 0 \} \Bigg)$$

I tried the it with simple functions like $f(x,y,z)=x^3+y^3+z^3$ to convince myself that it works but i don't really know how to prove it.

Any tipps or advice? Thanks in advance