Physical Applications of the Derivative

Average Velocity

The average velocity is the ratio of distance travelled (Δe) and the time elapsed (Δt).

Velocidad media

Instantaneous Velocity

The instantaneous velocity is the average speed limit when Δt approaches zero, that is to say, the derivative of space relative to time.

Velocidad instantánea

Instantaneous Acceleration

The instantaneous acceleration is the derivative of velocity relative to time.

Aceleración instantánea

Therefore, the acceleration is the second derivative of distance relative to time.

Aceleración instantánea


The distance travelled by a vehicle is given by the function e(t) = 3t² − t + 1. The distance is measured in meters and time in seconds.

Find the equation of velocity.

v(t) = e′(t) = 6t − 1

Find the velocity at time t = 0.

v(0)= 6 · 0 − 1 = −1 m/s

Find the acceleration equation.

a(t) = v′(t) = e′′(t) = 6 m/s2