  The depinning transition of a front moving in a time-independent random
potential is studied. The temporal development of the overall roughness w(L,t)
of an initially flat front, $w(t)\propto t^\beta$, is the classical means to
have access to the dynamic exponent. However, in the case of front propagation
in quenched disorder via extremal dynamics, we show that the initial increase
in front roughness implies an extra dependence over the system size which comes
from the fact that the activity is essentially localized in a narrow region of
space. We propose an analytic expression for the $\beta$ exponent and confirm
this for different models (crack front propagation, Edwards-Wilkinson model in
a quenched noise, ...).
