What is the derivative definition of instantaneous velocity?

Sep 9, 2014

Instantaneous velocity is the change in position over the change in time. Therefore, the derivative definition of instantaneous velocity is:

instantaneous velocity=
$v$= ${\lim}_{\Delta t \to 0}$$\frac{\Delta x}{\Delta t}$= $\frac{\mathrm{dx}}{\mathrm{dt}}$

So basically, instantaneous velocity is the derivative of the position function/equation of motion. For example, let's say you had a position function:

$x = 6 {t}^{2} + t + 12$

Since $v$=$\frac{\mathrm{dx}}{\mathrm{dt}}$, $v = \frac{d}{\mathrm{dt}} 6 {t}^{2} + t + 12 = 12 t + 1$

That is the function of the instantaneous velocity in this case. Note that it is a function because instantaneous velocity is variable- It is dependent on time, or the "instant." For every $t$, there is a different velocity at that given instant $t$.

Let's say we wanted to know the velocity at $t = 10$ and the position is measured in meters (m) while the time in measured in seconds (sec).

$v = 12 \left(10\right) + 1 = 121 \frac{m}{\sec}$