I had a previous question about the divergence in spherical coordinates and using the usual formula found on wikipedia "List of formulas in Riemannian geometry" I could not get the correct form of the divergence.
It turned out that I had to take the derivative with respect to "distance" and not with respect to the coordinates.
Could someone explain the intuition behind the difference between these two types of derivations, because I don't know when and why should I use each of them? Help please -- very confused.
I suppose that clear notion is not divergence, but de Rham differential. You do not need metrics at all to define de Rham operator. For details see
http://en.wikipedia.org/wiki/Divergence#Relation_with_the_exterior_derivative
Though de Rham operator do not depend on metric, the way how you identify 2-forms with vector vields and 3-forms with numbers do.
Note, that to define divergence you do not need metric itself, but rather volume form (namely you do need $g$, just $\operatorname{det}g$). So there can not be connection of derivative with respect to distance and divergence.
I even don't quite understand, what did you mean by "I had to take the derivative with respect to "distance" ". It is better to make this statement more formal if you still have a question. Maybe you have an observation about spherical coordinates, which do not hold in general.