On Christoffel and standard words and their derivatives

We introduce and study natural derivatives for Christoffel and standard words, as well as for characteristic Sturmian words. These derivatives, which are defined as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp., $ab^k$) and $b$ to $a^kb$ (resp.,$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are not just a single letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or of a standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.