Lyapunov's Direct Method

Question

The following non linear system is given: \begin{align} \dot{x}_1(t)&=x_2(t)\\ \dot{x}_2(t)&=-\frac{g}{l}\sin(x_1)-\frac{k}{m}x_2 \end{align} Is asked to apply the Lyapunov's direct method in order to show that the origin is stable. What I have done so far is to take the energy as Lyapunov function: $$\dot{V}(x)=mgl(1-\cos(x_1))+\frac{m}{2}l^2x_2^2$$ for which I have the following properties: $$V(0)=0$$ $$V(x)>0 \quad \forall x\neq0$$ The last property to check is that $\dot{V}(x)\leqslant0$ and, after some computations, I came out with this: $$\dot{V}(x)=-l^2kx_2^2$$ Now my question is: as the derivative is negative with respect to $x_2$ but does not depend on $x_1$, can I conclude the Lyapunov stability of the origin?