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 derivatives are:
$$\begin{align} \textbf{r} ' (t) &= \lim_{\Delta t \to 0} \frac{f(t + \Delta t)\textbf{i} + g(t + \Delta t)\textbf{j} - f(\Delta t)\textbf{i} - g(\Delta t)\textbf{j}}{\Delta t} \\ \textbf{u} ' (t) &= \lim_{\Delta t \to 0} \frac{h(t + \Delta t)\textbf{i} + k(t + \Delta t)\textbf{j} - h(\Delta t)\textbf{i} - k(\Delta t)\textbf{j}}{\Delta t} \end{align}$$
If we add them:
$$ \textbf{r}(t) + \textbf{u}(t) = f(t)\textbf{i} +h(t)\textbf{i}+ g(t)\textbf{j} + k(t)\textbf{j}$$
By definition of derivatives:
$$ \frac{d}{dt} [ \textbf{r}(t) + \textbf{u}(t) ] = \lim_{\Delta t \to 0} \frac{[f(t+\Delta t)\textbf{i} +h(t+\Delta t)\textbf{i}+ g(t+\Delta t)\textbf{j} + k(t+\Delta t)\textbf{j}]-[f(t)\textbf{i} +h(t)\textbf{i}+ g(t)\textbf{j} + k(t)\textbf{j}]}{\Delta t}$$
Rearraning:
$$ \frac{d}{dt} [ \textbf{r}(t) + \textbf{u}(t) ] = \lim_{\Delta t \to 0} \frac{[f(t+\Delta t)\textbf{i} + g(t+\Delta t)\textbf{j} -f(t)\textbf{i} - g(t)\textbf{j}] + [h(t+\Delta t)\textbf{i} + k(t+\Delta t)\textbf{j}-h(t)\textbf{i} - k(t)\textbf{j}]}{\Delta t}$$
According to the sum property of limits:
$$ \frac{d}{dt} [ \textbf{r}(t) + \textbf{u}(t) ] = \lim_{\Delta t \to 0} \frac{f(t+\Delta t)\textbf{i} + g(t+\Delta t)\textbf{j} -f(t)\textbf{i} - g(t)\textbf{j}}{\Delta t} + \lim_{\Delta t \to 0}\frac{h(t+\Delta t)\textbf{i} + k(t+\Delta t)\textbf{j}-h(t)\textbf{i} - k(t)\textbf{j}}{\Delta t}$$
By definition of a limit:
$$ \frac{d}{dt} [ \textbf{r}(t) + \textbf{u}(t) ] = \textbf{r}'(t) + \textbf{u}'(t) $$