Given 2 vector-valued functions u(t) and v(t), we have the product rule as follows.
$\frac{d}{d t}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\mathbf{u}^{\prime}(t) \cdot \mathbf{v}(t)+\mathbf{u}(t) \cdot \mathbf{v}^{\prime}(t)=\mathbf{u}^{\prime}(t) \mathbf{v^T}(t)+\mathbf{u^T}(t)\mathbf{v}^{\prime}(t)$
The notation $\cdot$, $T$ here are the dot product and transpose respectively.
The above dot product is equivalent to the multiplication of $u^T(t)$ and $v(t)$, which makes me wonder if we can have something similar for the multiplication of 2 vectors $u^T(t)=k(t)$ and $v(t)$?
$ \frac{d}{d t}[\mathbf{k}(t) \mathbf{v}(t)]= \mathbf{k}^{\prime}(t)\mathbf{v}(t)+\mathbf{k}(t) \mathbf{v}^{\prime}(t)$
In this case, k(t) is a row vector, as opposed to column vector $v(t)$.
Does it make any sense? If you have any reference for further reading, I would really appreciate.
Thank you!