 # Need help with some maths

Hi

I knew I should have kept awake during these classes, but sleeping was important back then What I want to know is at what point x a power function (x^n) is tangent to a 45° line.
So if I have i.e:
f(x) = x^2.6 (or any number),
at what point x does it “climb” at a 45° angle?

Sounds like what I’m looking for is when is the derivative = 1, right? Derivative would be (2.6*x)^(2.6-1), so I plug it into a solver and get this result, which in turns tells me I need to “rationalize decimal numbers”, whatever that means I found some code examples based on various languages, but I’m not much of a programmer. I see the basic versions seem to use loops, which doesn’t bide well for a Cycles approach.

The idea is to convert a linear input (known to be 1:1, or 45°) to something that is linear for a given range (x), and then switch to a power function with power n. I.e. no values change from 0-3.7 (or whatever), but from 3.7 onwards it looks like a power function that starts with an angle of 45° (but can be any power I choose).

Does anyone have an idea how to attack this?

Alternatively I could solve for a given number of useful presets of n and hardcode the output by a given n.

Edit: Oh my god, never mind. Apparently 1/n is exactly what I’m looking for (except in number format that makes sense), but feel free to correct me if I’m wrong.

The derivative is 2.6*x^1.6.
Then you do not need a solver for this, the solution is x=(1/2.6)^(1/1.6).

Yeah, it became obvious after a while. Still, I wasn’t able to complete a working node setup for a continuous curve (linear 1:1 up to some arbitrary point, then exponentially growing at a rate of own choosing, with tangential continuity of course). It was just an experiment, nothing important. Thanks anyway.