  Let $\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a
function $b$ in the unit ball of $H^\infty(\mathbb{C}_+)$. We study the
boundary behavior of the derivatives of functions in $\mathcal{H}(b)$ and
obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(\mu)} \le
C\|f\|_{\mathcal{H}(b)}$, where $f \in \mathcal{H}(b)$ and $\mu$ is a
Carleson-type measure on $\mathbb{C}_+\cup\mathbb{R}$. We provide several
applications of these inequalities. We apply them to obtain embedding theorems
for $\mathcal{H}(b)$ spaces. These results extend Cohn and Volberg--Treil
embedding theorems for the model (star-invariant) subspaces which are special
classes of de Branges--Rovnyak spaces. We also exploit the inequalities for the
derivatives to study stability of Riesz bases of reproducing kernels
$\{k^b_{\lambda_n}\}$ in $\mathcal{H}(b)$ under small perturbations of the
points $\lambda_n$.
