the relation between the set of derivated words of a Sturmian sequence $\uu$ and the set of derivated words of $\varphi(\uu)$, where $\varphi$ is a generator of the Sturmian monoid.

The main result of our article is an exact description of the morphisms fixing the derivated words $d_{\uu}(w) $ of $\uu$, where $\uu$ is fixed by a Sturmian morphism $\psi$ and $w$ is its prefix. For this purpose, we introduce an operation $\Delta$ acting on the set of Sturmian morphisms with unique fixed point, see . Iterating this operation we create the desired list of the morphisms as stated in . The Sturmian morphisms with two fixed points are treated separately, see .

We continue our study by counting the number of derivated words, in particular by counting the distinct elements in the sequence $\bigl(\Delta^k(\psi)\bigr)_{k \geq 1}$. This number depends on the decomposition of $\psi$ into the generators of the special Sturmian monoid, see below in .

Using this decomposition, provide the exact number of derivated words for two specific classes of Sturmian morphisms.

For a general Sturmian morphism $\psi$, gives a sharp upper bound on their number. The upper bound depends on the number of the elementary morphisms in 