I have understood it as follows: we want to find $D(\phi)_{(t,x)}:\Bbb{R}^2\rightarrow T_{\phi(t,x)}G$. So first we need to understand $T_{\phi(t,x)}G$. I know a basis for the lie algebra of $G$ which is $T_IG$ where the basis is $\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}, \begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}$. Now we can translate this basis to one of $T_{\phi(t,x)}G$. Which then is $\begin{pmatrix}
e^t & -xe^t\\
0 & -e^{-t}
\end{pmatrix}$, $\begin{pmatrix}
0 & e^t\\
0 & 0
\end{pmatrix}$. So we know that every element of $T_{\phi(t,x)}G$ can be written as $\lambda \begin{pmatrix}