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.

The key term here is second difference.  If $y=f(x)$ then the meaning of the symbol $d^2y$ is the infinitesimal version of the second difference $\Delta^2y=f(x)-2f(x+\Delta x)+f(x+2\Delta x)$.  If $f\in C^2$ then the second derivative $f''$ can be calculated in terms of the second difference by taking the limit of $\frac{\Delta^2 y}{\Delta x^2}$.  This explains Leibniz's notation for second derivative that the OP mentioned.
By far the clearest discussion of this is in Keisler's textbook Elementary Calculus, page 94.
