  Occurrences of a factor $w$ in an infinite uniformly recurrent sequence ${\bf
u}$ can be encoded by an infinite sequence over a finite alphabet. This
sequence is usually denoted ${\bf d_{\bf u}}(w)$ and called the derived
sequence to $w$ in ${\bf u}$. If $w$ is a prefix of a fixed point ${\bf u}$ of
a primitive substitution $\varphi$, then by Durand's result from 1998, the
derived sequence ${\bf d_{\bf u}}(w)$ is fixed by a primitive substitution
$\psi$ as well. For a non-prefix factor $w$, the derived sequence ${\bf d_{\bf
u}}(w)$ is fixed by a substitution only exceptionally. To study this phenomenon
we introduce a new notion: A finite set $M $ of substitutions is said to be
closed under derivation if the derived sequence ${\bf d_{\bf u}}(w)$ to any
factor $w$ of any fixed point ${\bf u}$ of $\varphi \in M$ is fixed by a
morphism $\psi \in M$. In our article we characterize the Sturmian
substitutions which belong to a set $M$ closed under derivation. The
characterization uses either the slope and the intercept of its fixed point or
its S-adic representation.
