# How does instantaneous rate of change differ from average rate of change?

Aug 4, 2014

Instantaneous rate of change is essentially the value of the derivative at a point; in other words, it is the slope of the line tangent to that point. Average rate of change is the slope of the secant line passing through two points; it gives the average rate of change across an interval.

Below is a graph showing a function, $f \left(x\right)$, and the secant line across an interval $\left[2 , 4\right]$. The slope of this secant line, which is

$\frac{\Delta y}{\Delta x} = \frac{f \left(4\right) - f \left(2\right)}{4 - 2}$

is the average rate of change of $f \left(x\right)$.

Below is a graph showing the function $f \left(x\right) = {x}^{2}$, as well as the line tangent at $x = 2$. The slope of this line is:

$\frac{\mathrm{dy}}{\mathrm{dx}} = f ' \left(2\right) = 2 \cdot 2 = 4$,

and it is the instantaneous rate of change at the point $\left(2 , 4\right)$.