Proving the existence of right derivative Suppose we have a continuous function $f:[a,b]\to\mathbb{R}$. We know that $f$ is differentiable on $(a,b]$ and that there is a finite limit:
$$\lim_{x\to a^{+}}f^{\prime}(x)$$
Can we prove that $f$ has a right derivative in point $a$? 

$$f'_+(a)=\lim_{x \rightarrow a^+} \frac{f(x)-f(a)}{x-a}.$$
By continuity of $f$, we have here indefinite symbol $\frac{[0]}{[0]}$. Using de L'Hospital rule we obtain
$$f'_+(a)=\lim_{x \rightarrow a^+} f'(x).$$
(The last limit exists by assumption).
