Hi Micah,
This is something that I hadn't considered formally before, but I've played with similar ideas over the years, at least for the product-based derivative. Since you posted this a few days ago, I've been playing with the product based derivative, when I can find time. I want to get a better understanding, before I attempt to tackle the next higher hyperoperation based derivatives.
My initial reaction is that the product based derivative bears a strong resemblance to the so-called logarithmic derivative, and indeed, to a first order approximation, they are related. I've worked out some of the details with pencil and paper, but haven't yet taken the time to write up some formal TeX expressions.
My curiosity then is whether the next higher order derivative would resemble some sort of superlogarithmic derivative, at least to a first order approximation? I haven't even tried to work it out yet, so I could be way off track with my guess. Anyway, thanks for bringing this to my attention. More to ponder and try to learn from!
Edit: the "logarithmic derivative" isn't as exciting as it sounds. It's just a regular derivative, like we're all used to, but you take the logarithm first. I didn't want to convey the impression that it's a different type of derivative, even if it does have some cool properties. The product based derivative that Micah posted does at first glance appear to be a different type of derivative.
Edit 2: link to wikipedia article on logarithmic differentiation:
https://en.m.wikipedia.org/wiki/Logarith...rentiation
This is something that I hadn't considered formally before, but I've played with similar ideas over the years, at least for the product-based derivative. Since you posted this a few days ago, I've been playing with the product based derivative, when I can find time. I want to get a better understanding, before I attempt to tackle the next higher hyperoperation based derivatives.
My initial reaction is that the product based derivative bears a strong resemblance to the so-called logarithmic derivative, and indeed, to a first order approximation, they are related. I've worked out some of the details with pencil and paper, but haven't yet taken the time to write up some formal TeX expressions.
My curiosity then is whether the next higher order derivative would resemble some sort of superlogarithmic derivative, at least to a first order approximation? I haven't even tried to work it out yet, so I could be way off track with my guess. Anyway, thanks for bringing this to my attention. More to ponder and try to learn from!
Edit: the "logarithmic derivative" isn't as exciting as it sounds. It's just a regular derivative, like we're all used to, but you take the logarithm first. I didn't want to convey the impression that it's a different type of derivative, even if it does have some cool properties. The product based derivative that Micah posted does at first glance appear to be a different type of derivative.
Edit 2: link to wikipedia article on logarithmic differentiation:
https://en.m.wikipedia.org/wiki/Logarith...rentiation
~ Jay Daniel Fox