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In a graph of position vs. time, the instantaneous velocity at any given point ##p(x,t)## on the function ##x(t)## is the derivative of the function ##x(t)## with respect to time at that point.

The **derivative** of a function at any given point is simply the instantaneous rate of change of the function at that point. In the case of a graph of position (or distance) vs. time, that means that the derivative at a given point ##p_0(t_0, x_0)## is the instantaneous rate of change in position (accounting for “positive” and “negative” direction) with respect to time.

As an example, consider a linear distance function (that is, one which can be represented with a line as opposed to a curve). If this were a function of ##x## and ##y##, with ##y## as the dependent variable, then our function in slope-intercept form would take the form ##y=mx+b##, where ##m## is the slope and ##b## is the value of ##y## at ##x=0##. In this case, ##t## is our independent variable and ##x## is our dependent, so our linear function would take the form ##x(t) = mt+b##.

From algebra, we know that the slope of a line measures the number of units of change in the dependent variable for every single unit of change in the independent variable. Thus, in the line ##x(t) = 2t + 5##, for every one unit by which ##t## increases, ##x## increases by 2 units. If we were to, for example, assign units of seconds to ##t## and feet to ##x##, then every second that passed (that is, every increase of one second in ##t##), position (or distance) would increase by two feet (that is ##x## would increase by two feet)

Since our change in distance per unit of change in time will remain the same no matter our starting point ##(x_0,t_0)##, in this case we can be assured that our instantaneous velocity is the same throughout. Specifically, it is equal to ##m = 2##. Differentiating the function with respect to ##t## yields the same answer. Note that this is only identical to our **average velocity** throughout the function by design: for a non-linear function (such as ##x(t) = t^2##) this would not be the case, and we would need to use differentiation techniques to find the derivatives of such functions.

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