It sounds like that the lecturer is saying
for the example we're discussing here, it would represent the distance of the car from its starting position. this should make more sense when you consider the change in the distance with respect at time, ie the slope of the distance time graph, ie how much distance you are covering per unit time, is just the speed.
Wiki gives this definition of derivative.
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
Where "with respect to" is being used.
Is the lecturer saying "with respect at" or "with respect to"?