I am working on some research project involving partial differential equations with respect to the variable $u(x,t)$ and I am stuck with this expression:
$$\frac{\partial}{\partial x} \left(\frac{\mbox{d}x}{\mbox{d}t}\cdot u(x,t)\right),$$
where $x=x(t)$.
Is there something wrong with this expression; or is expression just the following?
$$\frac{\mbox{d}x}{\mbox{d}t}\cdot\frac{\partial u}{\partial x}$$
Any help is appreciated.