Then there exists a c in (a, b) for which ƒ'(c) = 0.

Rolle's theorem states the following: suppose ƒ is a function continuous on the closed interval and that the derivative ƒ' exists on (a, b). Therefore, we cannot take the derivative at the endpoints. In summation: The left sided and right sided limit must exist and be equal for the derivative to exist at a given point, and by nature such two sided limits are not possible if we can only approach a point from one side. Using this knowledge, we can see that although the limit will exist on the left side as we approach the rightmost endpoint, we cannot determine the value of the limit as we approach from the right because those values will not be included in the domain of the function f(x). Notice that the limit is not specified as being left or right sided, so by the definition of the limit, the left sided and right sided limits as h->0 must exist and be equal for the derivative to exist. We define the derivative as follows (also known as the difference quotient): It would seem like we could, but to understand why it's not possible you need to return to the definition of the derivative as a limit.

Khan academy ap calculus