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 algebraic definition of the second derivative of $y=f(x)$ is
$$
f''(x)=\frac{d\left[\frac{dy}{dx}\right]}{dx}
$$
This can be expanded using the quotient rule
$$
f''(x)=\frac{d^2y}{dx^2}-\frac{dy\ d^2x}{dx^3}
$$
Furthermore, if you desire to evaluate $d^2y/d^2x$ this can be found by taking algebraic differentials
$$
dy=f'(x)dx
$$
$$
d^2y=f''(x)dx^2+f'(x)d^2x
$$
And finally
$$
\frac{d^2y}{d^2x}=f''(x)\frac{dx^2}{d^2x}+f'(x)
$$
Unfortunately, it is not possible to place a value on this expression because the quantity $d^2x/dx^2$ cannot be evaluated. In order to obtain a usefull result, algebraic manipulation of the derivitive quantities must be done to cancel out all of the differential terms ($dx$, $dy$, $d^2x$, etc.), only then the expression can be evaluated.
