# What is the derivative definition of instantaneous velocity?

##### 1 Answer
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).

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