calculus made easy
y = x^2. If x grows,
x^2 grows. And if
x^2 grows, then
y grows. So, let
x grow a little bit bigger and become
x + dx. It follows that y will grow a bit bigger and become
y + dy.
So, we have:
y + dy = (x + dx)^2
y + dy = x2 + 2x·dx + (dx)^2
y = x^2 and
(dx)^2 is effectively zero, so:
dy = 2x·dx
dy/dx = 2x
And we can prove, in general, that for:
y = x^n
dy/dx = nx^(n−1)
That’s half of calculus in a nutshell. Of course, the formulae get more complicated, but we can use computers to do the mechanical solving these days.
The thing that gets left out. The thing that really matters, imo, is what
dy/dx = 0, then it’s a horizontal line. It’s a local minima. It’s a change of behaviour (when applied to physical systems). And it’s something that can be calculated for.
dy/dx is interesting. Pretty much always.
Everyone knows about dark matter these days. The way we solve for dark matter is based on the flat (
dy/dx = 0) rotational velocity graph of galaxies
dy/dx = 1, then the x and y are growing 1:1. It’s the tipping point.
dy/dx starts getting big, then things are starting to run away. Like the GBP after Brexit.
Another way to think of calculus is from a system’s behaviour and how its function might look to create the observed behaviour.
I think it’s pretty cool that Newton’s ideas are being used to further our understanding of machine learning. I wonder what he’d make of it.