Using $dx$ for $h$. Is it mathematically correct to write
$$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx},$$
rather than
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}?$$
If not, what is the difference?  If so, why isn't this notation used from the beginning?  My feeling for the latter is that it would align the derivative more with the inverse of the indefinite integral.
 A: We do have the notation $f'(x) = \frac{df}{dx}$, which as you say, "aligns the derivative more with the inverse of the integral".  However, in its usual usage, the particle $dx$ is not in itself a number (see this question for more on that), so using $dx$ like that in a limit is misleading. 
A better usage might be as follows:
$$
\frac{df}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x}
$$
We could also apply this to integrals:
$$
\int_a^b f(x)\,dx = \lim_{\Delta x \to 0} \sum_{k=1}^{N(\Delta x)}
f(x^*_k)\,\Delta x$$
A: No it doesn't matter, as pointed out what you write doesn't matter, as long as the concept gets across, $dx$, $f$, $john$ $\oplus$, you can use any of them, $h$ is just the common one and used because everyone recognises it but in certain context, to make clear, you must use other letters as you're doing more than one thing at once.
A: It would be more common to use $\delta x$, to make it clear that it's "a small quantity" rather than the infinitesimal $dx$. Do note that you can't necessarily just manipulate $dx$-the-infinitesimal in the obvious ways: for instance, $$\dfrac{\partial a}{\partial b}_c \dfrac{\partial b}{\partial c}_a \dfrac{\partial c}{\partial a}_b = -1$$ rather than the $1$ you might expect, so funky things can happen with infinitesimals.
