Let $f$ be differentiable on an interval $I$ and let $x_0$ be an interior point of $I$. Make precise the following statement and prove it: $$\lim_{J \to x_0} \frac{|f(J)|}{|J|} = |f '(x_0)|$$
using the definition of limits where $$\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = f'(x_0).$$