Given x is a position vector, how would we convert x to eulerian and get the velocity vector? I know $d/dt$ allows us to get the velocity vector, but I don't know how the $\exp$'s will drop. Can you break it down step by step?
$$\begin{align}\textbf x &= \left(x_0 \exp\left(2t^2\right), y_0 \exp\left(−t^2\right), z_0 \exp\left(−t^2\right)\right)\\\textbf u &= (4xt, −2yt, −2zt)\end{align}$$
You are given $x(t)=x_0e^{2t^2}$. Then $$ \dot x(t)=x_02(2t)e^{2t^2}=4x(t)t. $$ The other coordinates are similar.
If you want to make it a little more systematic, use the logarithmic derivative.