  When a growing interface belonging to the KPZ universality class is tilted
with average slope $m$, its average velocity increases in
$\frac{\Lambda}{2}\,m^2$, where $\Lambda$ is related to the nonlinear
coefficient $\lambda$ of the KPZ equation. Nevertheless, a necessary condition
for this association to hold true is that the mean square height-gradient
increases in $b\, m^2$ when the interface is tilted. For the continuous KPZ
equation $b = 1$ and the relation $\Lambda=\lambda$ is achieved. In this work,
we study the local fluctuations of the height gradient through an analysis of
the values of $b$. We show that, for 1-dimensional discrete KPZ models, $b$ has
a power-law dependence with the discretization step $s$ chosen to calculate the
height gradient and $b$ goes to $1$ as $s$ increases. Its power-law exponent
$\gamma_b$ matches the exponent associated with the finite-size corrections of
the interface average velocity, $\textit{i.e.}$ $\gamma_b=2(\zeta-1)$, where
$\zeta$ is the global roughness exponent. We also show how, for restricted
(unrestricted) growth models, the value of $b$ goes to $1$ from below (above)
as $s$ increases.
