The $n$ th quantum derivative ${\cal D}_nf(x)$ of the real-valued function $f$ is defined for each real non-zero $x$ as \[ \lim_{q\to 1}\frac{\displaystyle\sum\limits^n_{k=0}(-1)^k\left[\begin{array}{c}n\\k\end{array}\right]_qq^{(k-1)k/2}f(q^{n-k}x)}{q^{(n-1)n/2}(q-1)^nx^n}, \] where $\left[\begin{array}{c}n\\k\end{array}\right]_q$ the $q$ -binomial coefficient. If the $n$ th Peano derivative exists at $x$ , which is to say that if $f$ can be approximated by an $n$ th degree polynomial at the point $x$ , then it is not hard to see that ${\cal D}_nf(x)$ must also exist at that point. Consideration of the function $|1-x|$ at $x=1$ shows that the second quantum derivative is more general than the second Peano derivative. However, it can be shown that the existence of the $n$ th quantum derivative at each point of a set necessarily implies the existence of the $n$ th Peano derivative at almost every point of that set.