Let \(\textbf{r}(t)\) and \(\textbf{u}(t)\) be defined as follows:
$$\begin{align} \textbf{r}(t) &= f(t)\textbf{i} + g(t)\textbf{j} + h(t)\textbf{k} \\ \textbf{u}(t) &= l(t)\textbf{i} + m(t)\textbf{j}+ n(t)\textbf{k} \end{align}$$
The cross product is:
$$\begin{gather} \textbf{r}(t) \times \textbf{u}(t) \\ = [g(t)n(t)-m(t)h(t)]\textbf{i} - [f(t)n(t)-l(t)h(t)]\textbf{j}+ [f(t)m(t)-l(t)g(t)]\textbf{k} \\ = g(t)n(t)\textbf{i}-m(t)h(t)\textbf{i} - f(t)n(t)\textbf{j}+l(t)h(t)\textbf{j}+ f(t)m(t)\textbf{k}-l(t)g(t)\textbf{k}\end{gather}$$
Taking the derivative of this:
$$\begin{gather} g'(t)n(t)\textbf{i}+g(t)n'(t)\textbf{i} - m'(t)h(t)\textbf{i} - m(t)h'(t)\textbf{i} \\ - f'(t)n(t)\textbf{j} - f(t)n'(t)\textbf{j} + l'(t)h(t)\textbf{j} + l(t)h'(t)\textbf{j} \\ + f'(t)m(t)\textbf{k} + f(t)m'(t)\textbf{k} - l'(t)g(t)\textbf{k} - l(t)g'(t)\textbf{k} \end{gather}$$
Rearraning:
$$\begin{gather} g'(t)n(t)\textbf{i}- m(t)h'(t)\textbf{i} - f'(t)n(t)\textbf{j} + l(t)h'(t)\textbf{j} + f'(t)m(t)\textbf{k}- l(t)g'(t)\textbf{k}\\+ g(t)n'(t)\textbf{i} - m'(t)h(t)\textbf{i} - f(t)n'(t)\textbf{j} + l'(t)h(t)\textbf{j} + f(t)m'(t)\textbf{k} - l'(t)g(t)\textbf{k} \end{gather}$$
Or:
$$\begin{vmatrix}\textbf{i} & \textbf{j} & \textbf{k} \\ f'(t) & g'(t) & h'(t) \\ l(t) & m(t) & n(t) \end{vmatrix} + \begin{vmatrix}\textbf{i} & \textbf{j} & \textbf{k} \\ f(t) & g(t) & h(t) \\ l'(t) & m'(t) & n'(t) \end{vmatrix}$$
This means:
$$\frac{d}{dt} [\textbf{r}(t) \times \textbf{u}(t)] = (\textbf{r}'(t) \times \textbf{u}(t)) + (\textbf{r}(t) \times \textbf{u}'(t)) $$