Extended real number N is a derived number at $x\in [0,1]$ of a real function f on [0,1] if there is a sequence $x_n \to x$, $x_n \in [0,1]\setminus \{x\}$ such that $ \frac{ f(x_n) - f(x) }{ x_n - x } \to N . $
[So this is something like a derivative, but you allow many of them since for each of them you have the above requirement only for a single sequence $x_n$.]
A special case is $ \liminf_{y\to x+} \frac{f(y)-f(x)}{y-x} $ which might be called right lower derived number of $f$ at $x$. [ Indeed you might find a sequence $x_n$ that realizes this number. ] Three more special cases are left and/or upper variants of this (That is, with "-" in place of "+", and/or with $\limsup$ in place of $\liminf$.
This should be defined e.g. in: V. Jarník: Über die Differenzierbarkeit stetiger Funktionen, Fund. Math. 21 (1933), 48-58.
The modern English literature I have to hand unfortunately only defines more advanced, stronger variants like essential derived number (starting with French V. Jarník: Sur les nombres dérivés approximatifs, Fund. Math. 22 (1934), 4—16).