Why rate of expansion of the universe is accelerating?
The accelerating expansion of the universe is caused (at least insofar as our current understanding goes) by the presence of dark energy. Dark energy has huge negative pressure, which has two important consequences.
First… when gravity works on a normal substance, it compresses it; if gravity were somehow removed, the compressed gas would expand, releasing the stored energy. Because dark energy has negative pressure, it actually expands under gravity. (Not as exotic as it sounds; fundamentally, it is no different from, say, a bubble “falling up” in the sea in a gravitational field, as a result of gravity.)
The second thing about dark energy is that all the work that gravity does when it makes it expand ends up as more dark energy. So the dark energy density never goes down, even as the energy density of all the other stuff that’s in the universe does go down as stuff get diluted.
So if the universe expands sufficiently, one day dark energy becomes the dominant component. And when that happens, its repulsive response to gravity makes the expansion speed up from that point on.
In our universe, this happened about 4–5 billion years ago. Since then, the expansion has been accelerating slowly, but ultimately, exponentially. Which means (in answer to the question details) that indeed, all higher derivatives of the scale factor of the universe will be positive, too.