q-derivative

Remarks

It is a difference quotient of $f$ between $x$ and $qx$. Example: for an integer $n \ge 1$ and $f(x) = x^n$ on $\R$, one gets $D_q x^n = \frac{q^n - 1}{q - 1} x^{n-1} = [n]_q \, x^{n-1}$, where $[n]_q = 1 + q + \cdots + q^{n-1}$ is the $q$-analog of $n$. As $q \to 1$, $[n]_q \to n$, and more generally $D_q f(x) \to f'(x)$ at each $x \ne 0$ where $f$ is differentiable. Non-example: with $q = 2$ and $f$ defined only on $A = [1, 2]$, the point $qx$ lies outside $A$ for $x > 1$, so $D_2 f(x)$ is not defined there. The operator is linear and satisfies the product rule $D_q (fg)(x) = f(qx) \, D_q g(x) + g(x) \, D_q f(x)$.