Jacobson radicals of Ore extensions

Let $R$ be a ring, $σ$ be an automorphism of $R$, and $D$ be a $σ$-derivation on $R$ with $Dσ=σD$. We will show that $J(R[x;σ,D])\subseteq A[x;σ,D]$ where $A$ is a collection of all leading coefficients of polynomials in $J(R[x;σ,D])$. In addition, we will prove that $J(R[x;σ,D])\cap R$ is nil if