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:

q^{dx-1}f(x) ≡ lim_{h→0}(f(x+h)/f(x))^{h-1} = exp(d/dx ln f(x))

For example, let's take e^{x}. How fast is the __ratio__ of successive values changing (it should be e, right?)

q^{dx-1}e^{x} = exp(d/dx ln e^{x}) = exp(1) = e.

As expected.

How fast is the __ratio__ of successive values changing in f(x) = x? This is a bit more interesting:

q^{dx-1}x = exp(d/dx ln x) = exp(x^{-1})

Cool, huh?

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(a_{1}) + … + ln(a_{n}) = ln(a_{1}a_{2}…a_{n})

Just take e to that power, and you have a product. This concept can be extended to the integral.