Part 5/5:
The proof hinges on the limit definition of derivatives, extended component-wise.
In practice, this means that for a vector function (\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}), the derivative is simply:
[
\boxed{\mathbf{r'}(t) = f'(t)\mathbf{i} + g'(t)\mathbf{j} + h'(t)\mathbf{k}}
]
This straightforward approach underscores the importance of understanding component-wise differentiation as a powerful technique in vector calculus.