Bounded point derivations on certain function spaces

Let $ X$ be a compact nowhere dense subset of the complex plane $ \mathbb{C}$, and let $ dA$ denote two-dimensional Lebesgue (or area) measure in $ \mathbb{C}$. Denote by $ \mathcal {R}(X)$ the set of all rational functions having no poles on $ X$, and by $ R^p(X)$ the closure of $ \mathcal {R}(X)$ in $ L^p(X,dA)$ whenever $ 1\leq p<\infty $. The purpose of this paper is to study the relationship between bounded derivations on $ R^p(X)$ and the existence of approximate derivatives provided $ 2<p<\infty $, and to draw attention to an anomaly that occurs when $ p=2$.