thumb|The mean value theorem: For each differentiable function with there is a with . The mean value theorem gives a relationship between values of the derivative and values of the original function. If is a real-valued function and and are numbers with , then the mean value theorem says that under mild hypotheses, the slope between the two points and is equal to the slope of the tangent line to at some point between and . In other words,
thumb|The mean value theorem: For each differentiable function with there is a with . The mean value theorem gives a relationship between values of the derivative and values of the original function. If is a real-valued function and and are numbers with , then the mean value theorem says that under mild hypotheses, the slope between the two points and is equal to the slope of the tangent line to at some point between and . In other words,