  We introduce and study natural derivatives for Christoffel and finite
standard words, as well as for characteristic Sturmian words. These
derivatives, which are realized 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
$a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are
longer than one letter, higher-order derivatives are naturally obtained. We
define the depth of a Christoffel or 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.
