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.

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.
