Let R be a ring with an endomorphism $sigma$ and a derivation $delta$. An ideal I of R is ($sigma,;delta$)-ideal of R if $sigma(I){subseteq}I$ and $delta(I){subseteq}I$. An ideal P of R is a ($sigma,;delta$)-prime ideal of R if P(${ eq}R$) is a ($sigma,;delta$)-ideal and for ($sigma,;delta$)-ideals I and J of R, $IJ{subseteq}P$ implies that $I{subseteq}P$ or $J{subseteq}P$. An ideal Q of R is ($sigma,;delta$)-semiprime ideal of R if Q is a ($sigma,;delta$)-ideal and for ($sigma,;delta$)-ideal I of R, $I^2{subseteq}Q$ implies that $I{subseteq}Q$. The ($sigma,;delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($sigma,;delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(sigma,delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(sigma,delta)}(R)$ is the smallest ($sigma,;delta$)-semiprime ideal of R; (2) For every extended endomorphism $ar{sigma}$ of $sigma$, the $ar{sigma}$-prime radical of an Ore extension $P(R[x;sigma,delta])$ is equal to $P_{sigma,delta}(R)[x;sigma,delta]$.