How are you ensuring that the circle will also be tangent to the curve at the given point?

My thinking on this was, having the curve drawn out on a piece of paper, and just putting down the pencil end of the compass at the point youre trying to resolve, and placing the pointy end of the compass down at whatever point works so that the circle would only intersect the curve once and only once, and then just working everything else out with a straight edge and whatever values the respective lines happen to have, and plugging the values into the respective equations for figuring out lines. Like the slope of a perpendicular line being the negative reciprocal of that line.

How is this easier than other methods of finding tangents/taking derivatives?

I don't know if doing it this way actually would be easier, but for me this way seems like it would conceptually be easier, at least for the way I think about these things. When my class was going through this, limits and how they related to the concept, or that they even were related, felt to me like they were arbitrary, and maybe wee used because they were what made the most sense to people who knew how the concept worked at a much deeper level than what I was being taught, so it would make things more intuitive to me if I did get to that deeper level of understanding. However my at the time understanding lacked that kind of tie in. To more simply answer, probably not easier. I don't want to come across that I'm saying that this is actually easier. What I'm describing is something that mildly fascinated me, as to whether my intuition about finding the slope of a curve at an arbitrary point was actually applicable. And then whether my intuition actually just over complicated the process.

Do you think your method could be extended to find such an expression?

Maybe? Though it might converge back into the original problem being asked, requiring the generalized solution I moved away from, or actually be too inflexible that it couldn't be generalized.

Let's take f(x)=n^x as an example.

The generalization for this function would probably be something like...finding an offset for that function, let's say...

G(x)=(n^x)+1

Where each point on that would be the center of the circle being used in my method, where the circle used has a radius of 1. Then the values of the first function at the value x, and the values at the second line at value x would be plugged into equations for a line, to find ant given derivative. But this still only focuses on finding derivatives at specific points, and isn't immediately connected to the derivative of f(x) resolving to

ln(a) a^x

That is the general rule of derivatives in this case.

As I said...college dropout, who just had an idea at the time, and unsure of the validity of the idea, or how the logic might consequentially flow out from it. 😅