What does it mean when you use the "d" in a derivative as a variable? It's been over 15 years since I last did anything with calculus at school, and I've forgotten most things, so please be gentle with your answers. :)
One thing that has always bugged me when getting into more advanced stuff was that the derivative simbols are suddenly manipulated in a way that doesn't make sense to me. For example, let's take the simple equation
$$y=12x^2+24x+45$$
An alternative notation for the same thing would be:
$$f(x)=12x^2+24x+45$$
Here we merely substituted $y$ for $f(x)$. It means the same thing.
If memory serves me right, when we take the first derivative of this function, we have three ways we can write it:
$$y'=24x+24$$
$$f'(x)=24x+24$$
$$\frac{dy}{dx}=24x+24$$
Now, it's this third notation that I'm talking about here. To my understanding the $\frac{dy}{dx}$ doesn't really mean anything by itself. It's just another way to specify that we're talking about a derivative of a function, just like the $'$ in the other two notations.
But then sometimes I come across an equation like this:
$$m\frac{\text{d}^2\mathbf{r}}{\text{d}t^2} = \frac{kqq'}{|\mathbf{r}|^2},$$
And here suddenly the $d$ is being treated like a variable, not to mention the $t$ and... And if memory serves me right, I've seen ever more extreme examples where the fraction $\frac{dy}{dx}$ itself is taken apart and each of the components used as a variable. I think this was popular back in university when I was talking course on differential equations. Barely passed that, half of it didn't make sense, and this was a large part of the reason why. I don't remember why I never asked my teacher about it back then.
So... what does it mean when we start to break the $\frac{dy}{dx}$ apart?
 A: This just means the 2nd derivative of $r$ with respect to $t$
$$\frac{\text{d}^2\mathbf{r}}{\text{d}t^2}$$
So it's again just a notation. It has always looked confusing to me too. But that's what it is, just a notation. The $d$ is not treated like a variable. Even though you see $d^2$ there you can't e.g. take square root of it (and get $|d|$ as a result), or anything like that.
See also:
https://en.wikipedia.org/wiki/Second_derivative
A: Rather then consider the $\operatorname{d}$ alone, one should view the expression $\operatorname{d}\!y/\operatorname{d}\!x$ as the following:
$$ \frac{\operatorname{d}\!y}{\operatorname{d}\!x} = \frac{\operatorname{d}\!}{\operatorname{d}\!x} y, $$
that is $ \operatorname{d}/\operatorname{d}\!x$ is the differential operator that applies to the function $y$.
In view of this, we have the second derivative as
$$ \frac{\operatorname{d}\!}{\operatorname{d}\!x} \frac{\operatorname{d}\!}{\operatorname{d}\!x}y = \frac{\operatorname{d}^2}{\operatorname{d}\!x^2}y = \frac{\operatorname{d}^2\!y}{\operatorname{d}\!x^2}. $$
A: Regarding the splitting of the $dy/dx$ thingie ...
Suppose we have
$$
y = 2x + 5
$$
Then, as you say, we can write
$$
y' = 2
$$
or
$$
\frac{dy}{dx} = 2
$$
Here, the $dy/dx$ just means the same as $y'$ -- it's the derivative of $y$ with respect to $x$. It's a single indivisible symbol, not really a fraction. It tells us that the rate of change of $y$ with respect to $x$ is equal to $2$. In other words, $y$ is changing twice as fast as $x$ is. So far, so good.
But then some people would feel at liberty to rearrange it, as if it were a fraction, and they'd write
$$
dy = 2\,dx
$$
In my view, this is a bit sloppy, but you can make sense out of it. It says that if you change $x$ by a small amount $dx$, then $y$ will change by an amount $dy$ that's twice as large. That's certainly consistent with the earlier statement that $y$ is changing twice as fast as $x$.
I personally think it's confusing to use $dy$ and $dx$ to denote small changes in this way. I'd much prefer to use $\delta$ instead of $d$ for this purpose, so I would write
$$
\delta y = 2 \, \delta x
$$
This makes exactly the same statement about the sizes of small changes, but it avoids the confusion with the $dx$ and $dy$ used earlier.
A: There are a variety of opinions on this.  The thing to keep in mind is that d is not a variable, but an operator.  Therefore, $dy$ is a shorthand for $d(y)$, and means "the differential of $y$".  When you add in the superscript numbers, in the case of differentials, it means we are applying the operator that number of times.  So, $d^2y = d(d(y))$.
You really can split the fraction $\frac{dy}{dx}$.  Infinitesimals are valid mathematics, and differentials represent infinitesimal values.
You can't split the fraction $\frac{d^2y}{dx^2}$, but that isn't the fault of the differentials, but of the notation.  If you want a notation for second derivatives where you can split the fraction, it is:
$$ y'' = \frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$$
This comes from just applying the quotient rule to the first derivative (i.e., actually treating the first derivative as a quotient).  You can get higher-order derivatives by continuing the process.
