> At what 'level' of maths or understanding do you reckon someone has to be in order to make sense of your introductory paragraph?
It varies from country to country, curriculum to curriculum. I would expect any student studying A-Level in the UK (the last year of secondary school) would know most of this, but not necessarily all the connections between them
>> "I find that the most elegant way to understand exponential-related things"...
> Okay, I know what exponential increase means. Something grows by an exponent of something. 2, 4, 16, ...
For something to increase exponentially means that at each step it is multiplied by a factor (usually greater than 1). In your example the factor is 2.
So yes, you have a handle on what it means for something to grow exponentially.
>> "is by defining the exponential as the function satisfying f'=f (with suitable normalization)"...
> I don't know what an exponential is.
There is more than one way to define an exponential, and in part that's what makes them so important. they turn up in multiple contexts, and being able to think of them in different ways is valuable.
> A function, to me, is something I call as part of code, so I am not sure if I might be confusing the programmer and mathematician-definition of 'function'.
In mathematics a function is a "thing" that takes a value from some domain and returns a value in some (possibly the same, but not necessarily) domain. So "log" is a function that takes positive real numbers and returns a real number (although it's also defined in other contexts such as complex numbers and abstract groups).
Polynomials are functions, usually we think of them as taking a real number and returning a real number: y=4x^3-x+5. Polynomials can also take other "types" such as complex numbers, matrices, group elements, etc.
But basically a function takes a value and returns a value.
> I don't really know how to interpret f'=f in any way.
The notation f' is used (in some contexts) to refer to the derivative of the function f. Given a function f we can ask: At this location, how quickly is it changing? That defines another function, which we call the derivative. This is differential calculus.
Given a function y=f(x) the derivative is often written as dx/dy.
> I also don't really know how normalization plays into it. For me, normalization is something related to homogenising a series of things. Sometimes it can be related to normalising the volume of a bunch of mp3 files, or in other cases it can be turning a bunch of badly formatted phone numbers into a common format for a database.
So the exponential function y=e^x, where e is the constant 2.71828..., is the unique function with the property that its derivative is the function itself. But the function y=2^x has nearly the same property once you permit multiplication by an appropriate other constant. That's the re-normalisation.
So an exponential is a function which, up to multiplication by a constant, is its own derivative. Hence f'=f up to normalisation.
>> "In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc."
> I have often heard terms like "eigen"-something, but this always causes me trouble, because I end up parsing it in German ("eigen" means "self" in German). So now it is the "self"-function of something?
When you have a transformation, often it does different complicated things in different directions. But sometimes you can find a direction where the transformation is especially simple, such as just a scaling.
Take a square and stretch, rotate, and sheer it. When you've done so there will be a direction where the result is a simple scaling. That's called an eigen-vector, and the amount of scaling is the associated eigen-value. By knowing the eigen-vectors and eigen-values of a transformation you can know nearly everything about the transformation as a whole.
(Note: a rotation has to eigen-vectors when you limit yourself to the real plane, but does have complex eigen-values.)
> Okay, of the 'derivative'. Is a derivative the mathematician's way of saying "figuring out something you don't know by using a bunch of other things you do know"?
No. As mentioned earlier, the derivative of a function is the measure of how fast it's changing. Example: if y=x^2-3x+1, then at location x the function is changing at rate 2x-3. If you plot y against x, then at any point the slope of the curve will depond on x, but is given by the function dy/dx = 2x-3.
> I do not know what a linear operator is at all.
A "Linear Operator" is a thing L that transforms Xs into Yx, but has the properties that
* L(x0+x1) = L(x0) + L(x1)
* L(c.x) = c.L(X)
where c is any suitable constant, and x, x0, and x1 are the things that L transforms. Linear operators are the basis of all machine learning (and many other things)
> I also do not know what ODEs are.
Ordinary Differential Equation. These are equations where you express a relationship between a function and its derivatives. So the equation earlier : f' = f : is an ODE. It says that the function f is equal to its own derivative.
ODEs are important in fluid dynamics, because the way a fluid flows depends on pressure changes, which are expressed as derivatives.
The ODE f"=c.f is used to describe Simple Harmonic Motion.
I hope that helps, but you have no contact details in your profile, so I can't contact you to offer more information.