q-Derivative

The q-Analog of the Derivative, defined by

$$\left({d\over dx}\right)_q f(x)={f(x)-f(qx)\over x-qx}.$$

For example,

$$\left({d\over dx}\right)_q \sin x = {\sin x-\sin(qx)\over x-qx}$$

$$\left({d\over dx}\right)_q \ln x = {\ln x-\ln(qx)\over x-qx}={\ln\left({1\over q}\right)\over(1-q)x}$$

$$\left({d\over dx}\right)_q x^2 = {x^2-q^2x^2\over x-qx}=(1+q)x$$

$$\left({d\over dx}\right)_q x^3 = {x^3-q^3x^3\over x-qx}=(1+q+q^2)x^2.$$

In the Limit $q\to 1$, the $q$-derivative reduces to the usual Derivative.

See also Derivative