Well, it's a bit tricky one.
I am to give a, let's say, lecture concerning the basics of calculus (note: I'm just a student). My audience is, let's say, not very keen on maths (they're mostly physics students) and I know there's no point in even trying to make them understand the Cauchy "$\varepsilon-\delta$" definition of limit and so forth. Instead, I'm going to focus mainly on applications of derivatives and integrals in computing moments of inertia etc.
But here's my concern. In modern calculus, $\frac{dy}{dx}$ is no longer a ratio, for it's a operator. Yet physicists tend to manipulate with derivatives as they still were fractions (you know - $u=x^2\rightarrow du=2xdx\rightarrow xdx=\frac{du}{2}$). In most cases, though not entirely rigorous, such manipulations yelds correct results.
And here is my question. I have to introduce the definition $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.$$
How should I tell them, that while the upper is the modern and rigorous definition of a limit, manipulating with $dy$ and $dx$ mostly works, but can yeld some absurd results and thus we have to watch out? How to avoid them getting confused? How to cope with, frankly, treating derivatives in two ways? Pay attention that they are not maths students and there is no point in making them learn calculus in an entirely rigorous way...
I have some experience in teaching, but this is the topic I'm particularly concerned with and I'd really appreciate any suggestions.