# What is the relationship between the Average rate of change of a fuction and derivatives?

Sep 19, 2014

The average rate of change gives the slope of a secant line, but the instantaneous rate of change (the derivative) gives the slope of a tangent line.

Average rate of change:

$\frac{f \left(x + h\right) - f \left(x\right)}{h} = \frac{f \left(b\right) - f \left(a\right)}{b - a}$, where the interval is $\left[a , b\right]$

Instantaneous rate of change:

${\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

Also note that the average rate of change approximates the instantaneous rate of change over very short intervals.