Referring to this article, I'd like to compile two key formulas for the temperature coefficient of resistance.
First is the formula at zero temperature.
Next is the formula at a specific temperature.
To put it more simply, the reciprocal of the temperature coefficient of resistance in the second formula is equal to the inferred absolute zero temperature, $ t_o $. So the second formula can be interpreted as, if you want to get the temperature coefficient of resistance at a given temperature that isn't zero you have to add that value to the inferred absolute zero temperature.
I'd like to know, is there a derivation for these formulas? I mean, I totally get how you could define the first formula to just be a reciprocal of that quantity. But with the second, I don't get the intuition behind just adding in the denominator. I mean, it doesn't have to be a derivation but at least some intuition on why it is like that so it would help me remember it other than just straight up memorizing.