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:

$\sum _{s=0}^n\frac{n_s}{d_s}=\frac{{\displaystyle \sum _{s=0}^n{n}_s}}{{\displaystyle \sum _{s=0}^n}{{\displaystyle d}}_s}$

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:

$\sum _{s=0}^n\frac{n_s}{d_s}=\frac{{\displaystyle \sum _{s=0}^n\frac{n_s}{d_s}\prod _{z=0}^n{d}_z}}{{\displaystyle \prod _{z=0}^n{d}_z}}$

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.