I have heard about a generalization of the calculus named 'quantized calculus'. In this calculus the derivative is defined as
$ df= [F,f]=Ff-fF $
Here $ F(g(x))= \frac{i}{\pi}\int_{-\infty}^{\infty}dt \frac{g(t)-g(s)}{t-s}$. In any case if this is the 'quantized' derivative , how can one defined a 'quantized integral'? How can one recover the usual definition of derivative from this $ \frac{d}{dx} $?