The instantaneous speed is the magnitude of the derivative of the displacement which is s = f(t) = 4.4*t^2 evaluated at that particular time-point.
In a limit form it is f'(x) = lim h-> 0 [f(x+h) - f(x)]/h and represents the slope between a point and some infinitesimally small point just after it (this is why the limit is h->0 since it is close to zero but not exactly zero).
You can also represent the limit as
[f(b) - f(a)]/[b-a] as b->a when b > a
which is what you were getting at (except you didn't use the right limit).