Flow in $x$-direction and pressure driven velocity

So assume a pressure-driven, incompressible, and steady flow in $x$-direction between 2 inf. fixed surfaces.

Why should the partial derivative $frac{partial P}{partial x}$ in the Navier-Stokes Eqn. be constant?

My understanding is that:

1. not 0, since it’s driving the flow
2. constant, so there is no acceleration (steady)

But why is it that a constant $frac{partial P}{partial x}$ maintains the velocity?

Wouldn’t the pressure difference $frac{dp}{dt}$ at each $x$ point act a force on the flow and thus accelerate the flow?