Why do the authors ask to check this only in the case $X=\mathbb R^k$? Does it imply that it's not valid for arbitrary $X$? I think it is valid for any maniofold $X\subset R^N$ (only such manifolds are considered in this book). Because $dc_{t_0}(1)$ is always equal to the directional derivative of $c(t)$ at $t=t_0$ in the direction of the "vector" $1\in \mathbb R^1$.
