When learning to add fractions, many students naively think that:

$\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$

This can be stated more generally as:

$\sum _{s=0}^{n}\frac{{n}_{s}}{{d}_{s}}=\frac{\sum _{s=0}^{n}{n}_{s}}{\sum _{s=0}^{n}{d}_{s}}$

This isn’t correct, of course, the sum is correctly written as

$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$

, for three fractions:

$\frac{{n}_{1}}{{d}_{1}}+\frac{{n}_{2}}{{d}_{2}}+\frac{{n}_{3}}{{d}_{3}}=\frac{{n}_{1}{d}_{2}{d}_{3}+{d}_{1}{n}_{2}{d}_{3}+{d}_{1}{d}_{2}{n}_{3}}{{d}_{1}{d}_{2}{d}_{3}}$

and in general:

$\sum _{s=0}^{n}\frac{{n}_{s}}{{d}_{s}}=\frac{\sum _{s=0}^{n}\frac{{n}_{s}}{{d}_{s}}\prod _{z=0}^{n}{d}_{z}}{\prod _{z=0}^{n}{d}_{z}}$

What I find particularly interesting is how this expression looks very similar to a generalization of the product rule from introductory calculus.

$\frac{d}{dx}\prod _{s=0}^{n}{f}_{s}=\sum _{s=0}^{n}\frac{{f}_{s}^{‘}}{{f}_{s}}\prod _{z=0}^{n}{f}_{s}$

I feel like there is a connection here that I am not seeing. I’ve tried looking this up and have found no reference to these general forms. I’d love to see how other people can run with this.