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q-Derivative

The q-analog of the derivative, defined by

(d/(dx))_qf(x)=(f(x)-f(qx))/(x-qx).
(1)

For example,

(d/(dx))_qsinx=(sinx-sin(qx))/(x-qx)
(2)
(d/(dx))_qlnx=(lnx-ln(qx))/(x-qx)=(ln(1/q))/((1-q)x)
(3)
(d/(dx))_qx^2=(x^2-q^2x^2)/(x-qx)=(1+q)x
(4)
(d/(dx))_qx^3=(x^3-q^3x^3)/(x-qx)=(1+q+q^2)x^2.
(5)

In the limit q->1, the q-derivative reduces to the usual derivative.

See also

Derivative

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Cite this as:

Weisstein, Eric W. "q-Derivative." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-Derivative.html

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