  The future Euclid space satellite mission will offer an invaluable
opportunity to constrain modifications to Einstein's general relativity at
cosmic scales. We focus on modified gravity models characterised, at linear
scales, by a scale-independent growth of perturbations while featuring
different testable types of derivative screening mechanisms at smaller
non-linear scales. We considered three specific models, namely JBD, a
scalar-tensor theory with a flat potential, the nDGP gravity, a braneworld
model in which our Universe is a four-dimensional brane embedded in a
five-dimensional Minkowski space-time, and $k$-mouflage (KM) gravity, an
extension of $k$-essence scenarios with a universal coupling of the scalar
field to matter. In preparation for real data, we provide forecasts from
spectroscopic and photometric primary probes by Euclid on the cosmological
parameters and the additional parameters of the models, respectively,
$\omega_{\rm BD}$, $\Omega_{\rm rc}$ and $\epsilon_{2,0}$. The forecast
analysis employs the Fisher matrix method applied to weak lensing (WL);
photometric galaxy clustering (GCph), spectroscopic galaxy clustering (GCsp)
and the cross-correlation (XC) between GCph and WL. In an optimistic setting at
68.3\% confidence interval, we find the following percentage relative errors
with Euclid alone: for $\log_{10}{\omega_{\rm BD}}$, with a fiducial value of
$\omega_{\rm BD}=800$, 27.1\% using GCsp alone, 3.6\% using GCph+WL+XC and
3.2\% using GCph+WL+XC+GCsp; for $\log_{10}{\Omega_{\rm rc}}$, with a fiducial
value of $\Omega_{\rm rc}=0.25$, we find 93.4\%, 20\% and 15\% respectively;
and finally, for $\epsilon_{2,0}=-0.04$, we find 3.4\%, 0.15\%, and 0.14\%.
(abridged)
