Chapter 4.8, Problem 40ES

Use the Mean-Value Theorem to prove the following result: Let $f$ be continuous at $x_0$ and suppose that ${\lim }_{x\to x_0}{f}^{\prime }\left(x\right)$ exists. Then $f$ is differentiable at $x_0,$ and $f^{\prime }\left(x_0\right)=\underset{x\to x_0}{\lim }{f}^{\prime }\left(x\right)$