Let \(\textbf{r}(t)\) and \(\textbf{u}(t)\) be defined as follows:
$$\begin{align} \textbf{r}(t) &= f(t)\textbf{i} + g(t)\textbf{j} \\ \textbf{u}(t) &= h(t)\textbf{i} + k(t)\textbf{j} \end{align}$$
The dot product is:
$$\textbf{r}(t) \cdot \textbf{u}(t) = f(t)h(t) + g(t)k(t)$$
The derivative of \(\textbf{r}(t) \cdot \textbf{u}(t)\) is:
$$\begin{align}\textbf{r}(t) \cdot \textbf{u}(t) &= \frac{d}{dt} [f(t)h(t) + g(t)k(t)] \\ &= [f'(t)h(t) +f(t)h'(t) + g'(t)k(t) + g(t)k'(t)] \end{align}$$
Rearraning:
$$\textbf{r}(t) \cdot \textbf{u}(t) = [f'(t)h(t)+ g'(t)k(t)]+[f(t)h'(t) + g(t)k'(t)]$$
According to the definition of a dot product:
$$\begin{align} \textbf{r}(t) \cdot \textbf{u}(t) &= [\langle f'(t), g'(t) \rangle \cdot \langle h(t),k(t) \rangle]+[\langle f(t), g(t) \rangle \cdot \langle h'(t),k'(t) \rangle] \\ &= [\textbf{r}'(t) \cdot \textbf{u}(t)]+[\textbf{r}(t) \cdot \textbf{u}'(t)] \end{align}$$