in my book I have this:
Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f \in C^1(\mathbb{R})$, and $x_0$ a point in which $f'(x_0) \ne 0$
Then
$\lim_{\delta\to0}$ $\frac{\mu( f([x_0 - \delta, x_0 + \delta ] )}{\mu( [x_0 - \delta, x_0 + \delta ])} = |f'(x_0)| $
Where $\mu$ is the Lebesgue measure.
I feel pretty OK with that considering that the first derivative should measure the rate of change of the function in relation with the rate of change of the independent variable but why that's the exact expression formalizing this concept?
This is almost literally just a rephrasing of the usual definition of the derivative. If $f'(x_0)\neq 0$, then $f$ is monotonic in a neighborhood of $x_0$; let's assume it's increasing. Then $f([x_0-\delta,x_0+\delta])$ is just the interval $[f(x_0-\delta),f(x_0+\delta)]$, so we are taking the limit of $$\frac{\mu([f(x_0-\delta),f(x_0+\delta)])}{\mu([x_0-\delta,x_0+\delta])}=\frac{f(x_0+\delta)-f(x_0-\delta)}{2\delta}.$$ This is just like the difference quotient defining $f'(x_0)$, except that we are taking a "two-sided" difference where we compare $f(x_0+\delta)$ and $f(x_0-\delta)$ instead of comparing each of them to $f(x_0)$. Indeed, we can break this fraction up as $$\frac{f(x_0+\delta)-f(x_0-\delta)}{2\delta}=\frac{1}{2}\left(\frac{f(x_0+\delta)-f(x_0)}{\delta}+\frac{f(x_0-\delta)-f(x_0)}{-\delta}\right)$$ which is just an average of two difference quotients (one for $h=\delta$ and one for $h=-\delta$). So, since each difference quotient is converging to $f'(x_0)$, so is our fraction.
(If $f$ is decreasing instead of increasing, we get essentially the same thing except with a minus sign since $f$ reverses the order of the endpoints of our interval, so we get $-f'(x_0)=|f'(x_0)|$.)