  Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras and $\theta$ be a
nonzero character on $\mathcal{A}$. Then the \textit{Lau product Banach
algebra} $\mathcal{A}\times_{\theta}\mathcal{U}$ associated with the Banach
algebras $\mathcal{A}$ and $\mathcal{U}$ is the $l^1$-direct sum
$\mathcal{A}\oplus\mathcal{U}$ equipped with the algebra multiplication
$(a,u)(a',u')=(ab,\theta(a)u'+\theta(a')u+uu')\,
(a,a'\in\mathcal{A},u,u'\in\mathcal{U})$ and $l^1$-norm. In this paper we shall
investigate the derivations and multipliers from this Banach algebras and study
the automatic continuity of these mappings. We also study continuity of the
derivations for some special cases of Banach algebra $\mathcal{U}$ and Banach
$\mathcal{A}\times_{\theta}\mathcal{U}$-bimodule $\mathcal{X}$ and establish
various results on the continuity of derivations and give some examples.
