  Let $K$ be a field of characteristic zero, $R = K[X_1,\ldots,X_n]$. Let
$A_n(K) = K<X_1,\ldots,X_n, \partial_1, \ldots, \partial_n>$ be the $n^{th}$
Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by
giving $\deg X_i = \omega_i $ and $\deg \partial_i = -\omega_i$ for $i
=1,\ldots,n$ (here $\omega_i$ are positive integers). Set $\omega =
\sum_{k=1}^{n}\omega_k$. Let $I$ be a graded ideal in $R$. By a result due to
Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic
$A_n(K)$-modules for each $i \geq 0$. In this article we compute the de Rham
cohomology modules $H^j(\mathbb{\partial}; H^1_{(f)}(R))$ for $j \leq n-2$ when
$V(f)$ is a smooth hypersurface in $\mathbb{P}^n$ (equivalently $A = R/(f)$ is
an isolated singularity).
