  Let $X$ be a compact subset of the complex plane. It is shown that if a point
$x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational
function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$ and if $X$
contains an interior cone, then the bounded point derivation can be represented
by the difference quotient if the limit is taken over a non-tangential ray to
$x_0$. A similar result is proven for higher order bounded point derivations.
These results extend a theorem of O'Farrell for $R(X)$, the closure of rational
functions with poles off $X$ in the uniform norm.
