In mathematics, in the area of combinatorics and quantum calculus, 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. For other forms of q-derivative, see Chung et al. (1994).
Definition The q-derivative of a function f(x) is defined as
( d d x ) q f ( x ) = f ( q x ) − f ( x ) q x − x . {\displaystyle \left({\frac {d}{dx}}\right)_{q}f(x)={\frac {f(qx)-f(x)}{qx-x}}.}
It is also often written as D q f ( x ) {\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
D q = 1 x q d d ( ln x ) − 1 q − 1 , {\displaystyle D_{q}={\frac {1}{x}}~{\frac {q^{d~~~ \over d(\ln x)}-1}{q-1}}~,}
which goes to the plain derivative, D q → d d x {\displaystyle D_{q}\to {\frac {d}{dx}}} as q → 1 {\displaystyle q\to 1} . It is manifestly linear,
D q ( f ( x ) + g ( x ) ) = D q f ( x ) + D q g ( x ) . {\displaystyle \displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~.}
It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms
D q ( f ( x ) g ( x ) ) = g ( x ) D q f ( x ) + f ( q x ) D q g ( x ) = g ( q x ) D q f ( x ) + f ( x ) D q g ( x ) . {\displaystyle \displaystyle D_{q}(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x)=g(qx)D_{q}f(x)+f(x)D_{q}g(x).}
Similarly, it satisfies a quotient rule,
D q ( f ( x ) / g ( x ) ) = g ( x ) D q f ( x ) − f ( x ) D q g ( x ) g ( q x ) g ( x ) , g ( x ) g ( q x ) ≠ 0. {\displaystyle \displaystyle D_{q}(f(x)/g(x))={\frac {g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(qx)g(x)}},\quad g(x)g(qx)\neq 0.}
There is also a rule similar to the chain rule for ordinary derivatives. Let g ( x ) = c x k {\displaystyle g(x)=cx^{k}} . Then
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