The velocity of a particle is the rate at which its distance in a straight line from a fixed point changes with time. Thus if we have the distance-time graph shown in Figure 22.1, then when the displacement changes by δx in a time δt the average velocity over that time interval is δx/δt. As St tends to 0 then δx/δt tends to a limiting value dx/dt which is the gradient of the tangent to the distance-time graph at a time t, point A on the graph, and so the instantaneous velocity v at the time t. Thus, since we have:

lim δt 0

δx δt =

dx dt

then the velocity at an instant of time is

v = dx dt

dx/dt will be positive if x increases as t increases and negative if x decreases as t increases.