Do two paths that merge into one have distinct derivatives at the "merge point"? Let $p_1, p_2: [a, b] \rightarrow \mathbb{R}^n$ be two distinct continuous and differential paths that merge at point $t$.  That is $p_1(t_x) = p_2(t_x)$ for all $t_x \ge t$ and $p_1(t_0) \ne p_2(t_0)$ for all $t_0 &lt; t$.
Does it necessarily follow that
$$
\frac{dp_1(t)}{dt} \ne \frac{dp_2(t)}{dt}
$$
or instead do we have equality
$$
\frac{dp_1(t)}{dt} = \frac{dp_2(t)}{dt}
$$
or is it indeterminate either way?
Assuming $t&lt;b$, then the derivative $p_1'(t)$ can be computed as the limit of $\frac{p_1(s)-p_1(t)}{s-t}$ as $s$ approaches $t$ from the right, and similarly for $p_2'(t)$.  Since $p_1(s)=p_2(s)$ for all $s\geq t$, these difference quotients are the same for $p_1$ and $p_2$, and so $p_1'(t)=p_2'(t)$.
Indeterminate.
Here is one where they are equal:
Curve 1:  $x=t, y=t^2$, when $t&lt;0,$ and $x=t,y=0, t\ge0.$
Curve 2:  $x=t, y=0, \forall t$
Here is a case where they are not equal:
Curve 1:  $x=t, y=t^2$, when $t&lt;1,$ and $x=t,y=1, t\ge1.$
Curve 2:  $x=t, y=2-t^2$, when $t&lt;1,$ and $x=t,y=1, t\ge1.$