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 . Definition The q-derivative of a function f(x) is defined as It is also often written as . 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 as . It is manifestly linear, It has a 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 . Then The eigenfunction of the q-derivative is the q-exponential eq(x). Relationship to ordinary derivatives Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is: where is the q-bracket of n. Note that so the ordinary derivative is regained in this limit. The n-th q-derivative of a function may be given as: provided that the ordinary n-th derivative of f exists at x = 0. Here, is the q-Pochhammer symbol, and is the q-factorial. If is analytic we can apply the Taylor formula to the definition of to get A q-analog of the Taylor expansion of a function about zero follows: Higher order q-derivatives Th following representation for higher order -derivatives is known: is the -binomial coefficient. By changing the order of summation as , we obtain the next formula: Higher order -derivatives are used to -Taylor formula and the -Rodrigues' formula (the formula used to construct -orthogonal polynomials). Generalizations Post Quantum Calculus Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator: Hahn difference Wolfgang Hahn introduced the following operator (Hahn difference): When this operator reduces to -derivative, and when it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems. β-derivative -derivative is an operator defined as follows: In the definition, is a given interval, and is any continuous function that strictly monotonically increases (i.e. ). When then this operator is -derivative, and when this operator is Hahn difference. Applications The q-calculus has been used in machine learning for designing stochastic activation functions. See also