Lyapunov function - second derivative

Question

I have a question concerning Lyapunov function, let's say that I have the first time derivative of Lyapunov function (V) and it is as follow: $\dot{V}(S)=-kS^2-\bar{k}|S|$. I need to derive a second time derivative and I am not sure if my solution is correct. Can someone confirm it or give me a hint how to get to the correct solution. $\ddot{V}(S)=-2kS\dot{S}-\frac{S}{|S|}\bar{k}\frac{S\dot{S}}{|S|}=-2kS\dot{S}-\bar{k}\frac{S^2}{|S|^2}\dot{S}=-2kS\dot{S}-\bar{k}\dot{S}$. Is it correct?

Answer 1

The derivative of |x| with respect to x is 1 if x> 0, -1 if x< 0, undefined at x= 0. That can, of course, be written as $\frac{x}{|x|}$. So the derivative of $\overline{k} |S|$ with respect to time is $\frac{S}{|S|}\dot{S}$. I don't know why you have $\frac{S}{|S|}$ twice. That is not correct.