Q-differentiation

In mathematics, in the area of combinatorics, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration.

The q-derivative of a function f(x) is defined as

It is also often written as ${\displaystyle D_{q}f(x)}$. The q-derivative is also known as the Jackson derivative.

Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator

which goes to the plain derivative, → ^d⁄_dx, as q → 1.

It is manifestly linear,

It has product rule analogous to the ordinary derivative product rule, with two equivalent forms

Similarly, it satisfies a quotient rule,

There is also a rule similar to the chain rule for ordinary derivatives. Let ${\displaystyle g(x)=cx^{k}}$. Then

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