# Can differentiation be used to find the average rate of change between two points rather than one?

I was wondering if I could use the derivative of a function to determine the average rate of change between two points, rather than one. I have a solution but I'm not sure whether it is valid or not: basically, to find the average rate of change between two points on any curve you must first know the rate of change at your maxima point, and subtract the rate of change of that point from the distance of it from your minima point in terms of x.

Thanks.

• You could just use a secant line that passes through both points. Commented Apr 11, 2015 at 14:53
• The derivative can only (directly) give an estimate of the slope of a secant line, not an exact value. Through the fundamental theorem you can use both differentiation and integration to get the true slope of a secant line.
– Ian
Commented Apr 11, 2015 at 14:54

The average rate of change in the interval $[a,b]$ is $$\frac{f(b)-f(a)}{b-a}$$ if the function is differentiable you can think about it like the sum of all of the derivatives in the specific interval divided by its length: $$\frac{1}{b-a} \int_a^b f'(t)dt = \frac{f(b)-f(a)}{b-a}$$