  We propose a differential hydrodynamic response relation, $V_2(\zeta)=\int
d\xi G(\zeta-\xi) \mathcal{E}_2(\xi)$, to describe the formation of a
pseudorapidity dependent elliptic flow in heavy-ion collisions, in response to
a fluctuating three-dimensional initial density profile. By analyzing the
medium expansion using event-by-event simulations of 3+1D MUSIC, with initial
conditions generated via the AMPT model, the differential response relation is
verified. Given the response relation, we are able to separate the two-point
correlation of elliptic flow in pseudorapidity into fluid response and
two-point correlation of initial eccentricity. The fluid response contains
information of the speed of sound and shear viscosity of the medium. From the
pseudorapidity dependent response relation, a finite radius of convergence of
hydrodynamic gradient expansion is obtained with respect to realistic fluids in
heavy-ion collisions.
