Derivation of stable spot price

[xenide]

I'm trying to ascertain the correctness of the function that calculates the stable spot price. Referring to the code here.

1. how is this equation derived?
2. why is that S (sum of balances) is 0?

[xenide]

paging @nventuro, could I trouble you take a quick look at this?

[jubeira]

Hey @xenide,

We've never really used this code in production, and it's been 4 years since its implementation.
The spot price is the slope of the invariant function evaluated at the current pool balances, so I assume the formula comes from deriving the implicit invariant function for stable pools. I can't really assure it's correctness at this point though.