# What is the derivative definition of instantaneous velocity?

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

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

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The derivative definition of instantaneous velocity is the rate of change of position with respect to time at a specific instant. Mathematically, it is expressed as the derivative of the position function ( s(t) ) with respect to time ( t ) at a particular time ( t_0 ). In symbols, the instantaneous velocity ( v(t_0) ) at time ( t_0 ) is given by:

[ v(t_0) = \lim_{\Delta t \to 0} \frac{s(t_0 + \Delta t) - s(t_0)}{\Delta t} ]

Where:

- ( v(t_0) ) is the instantaneous velocity at time ( t_0 ),
- ( s(t) ) is the position function representing the displacement of an object at time ( t ),
- ( \Delta t ) is a small interval of time,
- ( s(t_0 + \Delta t) - s(t_0) ) represents the change in position over the time interval ( \Delta t ),
- The limit as ( \Delta t ) approaches 0 ensures that the velocity is calculated at an infinitesimally small time interval around ( t_0 ), capturing the instantaneous rate of change of position with respect to time.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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