Let \(\textbf{r}(t)\) be defined as follows:
$$ \textbf{r}(t) = f(t)\textbf{i} + g(t)\textbf{j} + h(t)\textbf{k} $$
The derivative is:
$$ \frac{d}{dt} \textbf{r}(t) = f'(t)\textbf{i} + g'(t)\textbf{j} + h'(t)\textbf{k} $$
If we use \(m(t)\) instead of \(t\):
$$ \frac{d}{d \ m(t)} \textbf{r}(m(t)) = f'(m(t))\textbf{i} + g'(m(t))\textbf{j} + h'(m(t))\textbf{k} $$
If we want to derivative with respect to \(t\), we will have to use the chain rule:
$$\begin{align} \frac{d}{d t} \textbf{r}(m(t)) &= f'(m(t))m'(t)\textbf{i} + g'(m(t))m'(t)\textbf{j} + h'(m(t))m'(t)\textbf{k} \\ &= (f'(m(t))\textbf{i} + g'(m(t))\textbf{j} + h'(m(t))\textbf{k}) m'(t) \end{align}$$
This means:
$$ \frac{d}{d t} \textbf{r}(m(t)) = \textbf{r}'(m(t)) m'(t) $$