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Given an inner function $\Theta$ in the unit disc $\mathbb{D}$, we study theboundedness of the differentiation operator which acts from from the modelsubspace $K\_{\Theta}=\left(\Theta H^{2}\right)^{\perp}$ of the Hardy space$H^{2},$ equiped with the $BMOA$-norm, to some radial-weighted Bergman space.As an application, we generalize Peller's inequality for Besov norms ofrational functions $f$ of degree $n\geq1$ having no poles in the closed unitdisc $\overline{\mathbb{D}}$.

Boundedness of the differentiation operator in model spaces and application to Peller type inequalities