Time derivative of a function involving absolute value

Question

I have a functional looking like this $$ L[u] = \int \int_{\Omega} k \left| \nabla u \right|^2$$ , for which I want to take the time derivative $\dot{L}$. I am not sure that I handled the time derivation of the absolute function correctly, I did it this way - $$\dot{L}[u] = \int \int_{\Omega} k \frac{\partial}{\partial t}\left| \nabla u \right|^2 = \int \int_{\Omega} 2k \left| \nabla u \right| \left|\frac{\partial}{\partial t} \nabla u \right|= \int \int_{\Omega} 2k \left| \nabla u \right| \left|\nabla\frac{\partial}{\partial t} u \right|$$ Any math expert can confirm that I am doing the right thing here?