Consider
For example consider \begin{align*} \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} tx \\ y + t \end{pmatrix} . \end{align*} If I were to let \begin{align*} \mathbf{z} = \begin{pmatrix} x \\ y \\ t \end{pmatrix} \end{align*} then \begin{align*} \dot{\mathbf{z} } = \begin{pmatrix} t x \\ y + t \\ 1 \end{pmatrix} \end{align*} Then we have \begin{align*} \frac{\partial \dot{ \mathbf{z}} }{\partial t} = \begin{pmatrix} x \\ 1 \\ 0 \end{pmatrix} . \end{align*}
But this doesn't seem enitrely right. I think that now that $t$ is coordinate we should have \begin{align*} \frac{\partial \dot{ \mathbf{z}} }{\partial t} =\mathbf{0} . \end{align*}
Question: What is going with the the derivatives and which is which?