  An attempt has been made to explore the galactic dynamics via the rotational
velocity beyond the Einstein's geometric theory of gravity. It is inspired from
the geometric relation obtained in the power law $f(R)$ gravity model in
vacuum. We analyse the action with a small positive deviation from the
Einstein-Hilbert action (taking $R$ as $f(R)\propto R^{1+\delta}$) at the
galactic scales for the explanation of cosmological dark matter problem and
obtain the contribution of dynamical $f(R)$ background geometry in accelerating
the test mass. In the weak field limits, we obtain the effective acceleration
of the test mass due to a massive spherically symmetric source in $f(R)$
background and develop an equation for the rotational velocity. We test the
viability of the model by tracing the motion of test mass outside the typical
galactic visible boundaries without considering any dark matter halo profile.
We obtain a nice agreement in the outer regions (up to few tens of kpc beyond
the visible boundary) of the typical galaxy by using the known galaxy data.\\
We further explore the galactic dynamics for a galaxy NGC 1052 of which the
dark matter deficient galaxies, i.e., DF2 and DF4 are a part (satellite
galaxies) and discuss plots of the dynamical feature of rotation curves in
$f(R)$ background for the model parameter $\delta<<1$ and interpret the results
for its satellite galaxies.
