Reducing explanation of the derivative at an intuitive level to 'simply the rate of change' confuses the hell out of people when they encounter other things that are also defined as derivatives but do not describe a change in any obvious sense. For example, electric current (or, say, a flow of water) through a cross-section of a closed circuit does not necessarily represent a change of electic charge (or the mass of water) on either side of the surface, as it remains constant.
I think it is called Parametric Calculus.
v = u +at; v^2 = u^2 +2as;
and all that
Velocity = ds/dt and acceleration = dv/dt = d^2S/dt^2
At least in my mind rate of changes makes a lot of sense to explain derivatives.
edit: i.e Acceleration is the rate of change of velocity over time.