The product calculus is the study of how ratios between sucessive values in a mapping change, rather than differences. Really, it's just ordinary differential calculus with some logarithms thrown around.
The product derivative, the foundation of the study, is defined as:
qdx-1f(x) ≡ limh→0(f(x+h)/f(x))h-1 = exp(d/dx ln f(x))
For example, let's take ex. How fast is the ratio of successive values changing (it should be e, right?)
qdx-1ex = exp(d/dx ln ex) = exp(1) = e.
How fast is the ratio of successive values changing in f(x) = x? This is a bit more interesting:
qdx-1x = exp(d/dx ln x) = exp(x-1)
The product integral is defined in a similar way, so I'll just show the transformation:
∫f(x)dx = exp(∫ln f(x)dx).
A nice way to think about this is in terms of a sum.
ln(a1) + … + ln(an) = ln(a1a2…an)
Just take e to that power, and you have a product. This concept can be extended to the integral.