Average acceleration describes the change in velocity over a finite interval. Instantaneous acceleration describes the rate at which velocity changes at one particular time.

If velocity is $v(t)$, then

$$a(t)=\frac{dv}{dt}.$$

Because velocity is itself the derivative of position,

$$a(t)=\frac{d^2x}{dt^2}.$$

From average to instantaneous acceleration

Over a short interval $\Delta t$,

$$\frac{v(t+\Delta t)-v(t)}{\Delta t}$$

is an average acceleration. Letting the interval shrink to zero gives the derivative and therefore the instantaneous acceleration.

Velocity-time graph

On a velocity-time graph, instantaneous acceleration is the slope of the tangent line. A positive slope gives positive acceleration, a negative slope gives negative acceleration, and a horizontal tangent gives zero instantaneous acceleration.

Example

If

$$x(t)=t^3,$$

then

$$v(t)=3t^2$$

and

$$a(t)=6t.$$

At $t=2,\mathrm s$, the acceleration is $12,\mathrm{m/s^2}$ when position is measured in metres.

As with average acceleration, the sign of instantaneous acceleration must be interpreted together with the sign of velocity to determine whether speed is increasing or decreasing.