Differential of discount process

Question

I am studying financial mathematics and I am trying to show that if the discount process is $$ D(t)=\text{exp} \left[-\int_0^t R(s) ds\right] $$ then its differential is $$ dD(t)=-D(t)R(t)dt. $$ I think that this should be quite easy to show and I am guessing that Itô's lemma could be used. However, my book just states that this holds without showing it. What confuses me is that $D(t)$ depends on the path of $R(t)$. Am I still allowed to write $D(t)$ as $f(t, R(t))$ and then apply Itô's lemma?