How 2nd derivative is determined? Question is from string theory (A string model).

Question

[enter image description here][1]

This question is related to Mathematical modeling. I have many times to get second derivative but failed to drive. I would remain thankful for resolving my issue. [1]: https://i.stack.imgur.com/M4sA5.jpg

Answer 1

\begin{align*} \frac{D^2u}{Dt^2}&=\frac{D}{Dt}\left(\frac{Du}{Dt}\right) \\ &=\left(\frac{\partial}{\partial t}+V\frac{\partial}{\partial x}\right) \left(\frac{\partial u}{\partial t}+V\frac{\partial u}{\partial x}\right) \\ &=\frac{\partial^2u}{\partial t^2}+\frac{\partial V}{\partial t}\frac{\partial u}{\partial x}+V\frac{\partial^2 u}{\partial t \partial x}+V\frac{\partial^2 u}{\partial x \partial t} +V^2\frac{\partial^2 u}{\partial x^2} \\ &=u_{tt}+V_tu_x+2Vu_{xt}+V^2u_{xx} \end{align*}