This question is related to rough path theory. A link to wikipedia is: https://en.wikipedia.org/wiki/Rough_path
As I understand a signature determines a path. For this question consider a one dimensional smooth path and a corresponding signature. The question is: can there be relation between the signature obtained from the derivative of the path and the signature obtained from the original path. I understand the signature is not unique. Therefore, I am looking for a procedure or process to derive a signature so that signature from the derivative of the path can be derived from the original path/signature. Please accept my apologies if the question is not clear.
If $g(x)$ is smooth (assuming you mean at least $C^1$), then the iterated integrals are just Stieltjes integrals and
$$\int_s^t g(r)-g(s) dg (r)=\int_s^t (g(r)-g(s)) g'(r) dr$$
No rough path theory needed, because the path is smooth, lol.
Edit: for the updated question, no there is no nice relation as far as I know. For example take any path $g$ that is $C^{1+\alpha}$ for $\alpha<1/2$. I.e. $g$ is differentiable with $C^\alpha$ derivative. Then the signature of $g$ is just iterated Stieltjes integrals, but for $g'$ you need to postulate the iterated integrals which is of course non unique.
So no, there is no nice relationship.