The mean value theorem for derivatives states that at some point in a closed interval the average rate of change of a function is equal to the instantaneous rate of change at that point.

Formally the mean value theorem for derivatives is stated as follows

Given the function if is differentiable on the open interval and continuous on the closed interval .

Then there exists a point in such that the derivative at is equal to the average rate of change over the interval .

Alternatively,

For some value of such that

Example

Given the function find the location where the tangent line is equal to the average rate of change between and .

The function is depicted in red, with the secant line between and being shown in purple. The tangent line to is shown in orange.