# What is Quantum Calculus

Consider the following expression:

$\frac{f\left(x\right)-f\left({x}_{0}\right)}{x-{x}_{0}}$

As $x$ approaches ${x}_{0}$, the limit, if it exists, gives the familiar definition of the derivative $\frac{df}{dx}$ of a function $f\left(x\right)$ at $x={x}_{0}$. However, if we take $x=q{x}_{0}$. or $x={x}_{0}+h$. where $q$ is a fixed number different than $1$, and $h$ a fixed number different from $0$, and do not take the limit, we enter the fascinating world of quantum calculus: The corresponding expressions are the definitions of q-derivative and h-derivative of $f\left(x\right)$. Beginning with these two definitions [...] two types of quantum calculus, the q-calculus and h-calculus [are developed].

from "Quantum Calculus"

Quoted on Tue Nov 20th, 2012