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  "body": "\n## Summary\n\nThe $q$-derivative of a function $f$ is the quotient $\\frac{f(qx) - f(x)}{qx - x}$, where $q \\ne 1$ is a fixed number.\nIt replaces the limit in the ordinary [[derivative|0V1EJCT8G6]] by a comparison of $f$ at $x$ and at the scaled point $qx$.\nIt is the basic operator of $q$-calculus and of the theory of [[q-analogs|GFSDMJHHVA]], and it tends to the ordinary derivative as $q \\to 1$.\nIt is also called the Jackson derivative.\n\n## Explanations\n\n### L1 human\n\nThe ordinary derivative compares $f(x)$ with $f(x + h)$ for tiny $h$ and takes a limit.\nThe $q$-derivative compares $f(x)$ with $f(qx)$, a point that is a fixed factor away, and takes no limit at all.\nFor $f(x) = x^2$ it gives $\\frac{q^2 x^2 - x^2}{qx - x} = (1 + q) x$, which tends to the usual $2x$ as $q \\to 1$.\nThe number $1 + q$ in place of $2$ is a first example of a $q$-number.\n",
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