The Nature of Differentials and Infinitesimals I have been wondering for some time what the limits of Leibniz notation is, and what exactly its meaning is.  I learned limits and later learned (to some extent) infinitesimals, but there are some oddities which have me befuzzled.  The one person I know who could answer the question gave me a reference so dense I couldn't make heads or tails of it.
In any case, let's say you have a function $y = f(x)$.  Now, the derivative is $\frac{dy}{dx} = f'(x)$ and the second derivative is $\frac{d^2y}{dx^2} = f''(x)$. Anyway, if you play around with these a bit, you can see that $\frac{dx}{dx} = 1$, which means that $x$ always changes in unity with itself.  However, a very odd result happens if you look at the second derivative.  Since $\frac{dx}{dx} = 1$, and 1 is a constant, that means that the second derivative, $\frac{d^2x}{dx^2} = 0$, which means that x never has any acceleration with respect to itself.
However, algebraically, what this seems to mean to me is that $d^2x$ is always zero, but this is obviously not the case, as it could be put in ratio with $dy^2$ to produce a real-valued function.  However, this seems to be at odds with an infinitesimal definition of $d^2x$ (or any other definition I have seen).  It seems to imply that that $dx$ is more of a relational quantity than an infinitesimal or even a limit.
I did not know if anyone had any specific knowledge about this, or knew of any books that dealt with this topic.  I have a hard time finding any at all that approach this subject.
On a side note (but related), I would also be interested in any books which discussed any possible meaning of quantities like $\frac{d^2y}{d^2x}$ (note that this is different from the Leibniz second derivative which is $\frac{d^2y}{dx^2}$).  Anyway, if anyone has ideas or references, I would love to investigate this topic further.

$\frac{dx}{dx} = 1$ because the infinitesimal $dx$ changes at the same rate as $dx$.  
$\frac{d}{dx}\frac{dx}{dx} = 0$ because $\frac{dx}{dx}$ changes infinitely slower than $dx$.  One way to represent this is $\frac{d}{dx}\frac{dx}{dx} = lim_{n \rightarrow \infty} \frac{(\frac{1}{n})}{1} = 0$
The same holds true for $\frac{d^2 x}{dx^2}$.  It's not that $d^2 x = 0$, rather, it is that $dx^2$ changes infinitely faster than $d^2 x$.  Both are infinitesimals, so on their own they both approach zero.  But the ratio approaches zero.  

Some good reading is found on another page: Is There a Difference Between $d^2x$ and $(dx)^2$?
