I think, I've explained this already some days ago in this forum.
Again, the Euler coordinates \vec{x}_0 are labels for a fixed fluid element by noting its position at a fixed (initial) time t_0. Then you follow this fluid element along its trajectory in time. This is given by the position-vector field (as a function of the Euler coordinates) \vec{x}(t,\vec{x}_0).
In the Lagrange description you switch to a field description, i.e., you characterize the fluid by the velocity field, where an observer at point \vec{x} measures the velocity of the fluid element at this position at time, t. This field we denote by \vec{v}(t,\vec{x}).
The relation to the Euler coordinates is then given by
\vec{v}_0(t,\vec{x}_0)=\frac{\partial \vec{x}(t,\vec{x}_0)}{\partial t}=\vec{v}[t,\vec{x}(t,\vec{x}_0)].
It gives the velocity of the fluid element, which has been at position \vec{x}_0 at time t.
The same procedure holds for any other physical quantity which has a specific meaning for a given fluid element. An example is the temperature T. In the Euler description, the temperature of the fluid element which has been at \vec{x}_0 at time t_0 at time t is given by the function T_0(t,\vec{x}_0).
In the Lagrange description, you consider the temperature at the position \vec{x} of the fluid element that is there at time, t. This function is called T(t,\vec{x}). Again, the relation between the two points of view are given by
T_0(t,\vec{x}_0)=T[t,\vec{x}(t,\vec{x}_0)].
Now if you want the time derivative of the temperature of the specific fluid element that has been at position \vec{x}_0 at time t_0, you have to use the chain rule of multi-variable calculus,
\partial_t T_0(t,\vec{x}_0)=\left (\frac{\partial T(t,\vec{x})}{\partial t} \right)_{\vec{x}=\vec{x}(t,\vec{x}_0)}+ \frac{\partial \vec{x}(t,\vec{x}_0)}{\partial t} \cdot [\vec{\nabla} T(t,\vec{x})]_{\vec{x}=\vec{x}(t,\vec{x}_0)}=\left (\frac{\partial T(t,\vec{x})}{\partial t} \right)_{\vec{x}=\vec{x}(t,\vec{x}_0)}+\vec{v}[t,\vec{x}(t,\vec{x}_0)] \cdot [\vec{\nabla}\vec{\nabla} T(t,\vec{x})]_{\vec{x}=\vec{x}(t,\vec{x}_0)}.
I hope with this very detailed writing of all the arguments, the two (equivalent) descriptions of fluid mechanics and how to switch from one to the other has become more clear.
