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

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

See also Derivative

© 1996-9 Eric W. Weisstein