In my lecture notes, we have $$\dfrac{d\vec{x}}{dt}=f(\vec{x}),$$where $f$ is polynomial and $\vec{x}=0\in\mathbb{R}^n$ is an equilibrium.
In determining whether all trajectories $\vec{x}(t)\rightarrow 0$ as $t\rightarrow 0$, we want to find a Lyapunov function $V(\vec{x}(t))$ because $$\dfrac{d}{dt}V(\vec{x}(t))=\langle f(\vec{x}(t)),\ \nabla V(\vec{x}(t))\rangle.$$ I understand the derivation of that last formula, but I don't see how it helps determine whether all $\vec{x}(t)\rightarrow 0$ as $t\rightarrow 0$.
Can someone explain this?