Disclaimer - I'm not a mathematician, I'm a dirty physicist.
My work often involves performing calculus on various things without thinking about what I'm doing too much (I leave the proof of various identities etc for the pure mathematicians to worry about).
However I've often noticed that mathematicians get a little upset when I do tricks such as treating the differential $\frac{\text{d}y}{\text{d}x}$ as if it were a fraction.
The simplest example I can think of is how I think about the chain rule: $$\frac{\text{d}y}{\text{d}x} = \frac{\text{d}y}{\text{d}u}\frac{\text{d}u}{\text{d}x}$$ In my head, I imagine the $\text{d}u$ terms cancelling, which is why this works. Indeed, this is how I'll explain to others how to use the chain rule when asked about it.
My question is the following:
Is is dangerous to think about differentials in this way?
After all, one of the very first examples of calculus I've ever seen (back when I was a baby in high-school) was the derivative of $y=x^2$ calculated from first principles in the following way:
\begin{align} y&=x^2\\ y+\delta y&=(x+\delta x)^2\\ y+\delta y&=x^2+2x\delta x+\delta x^2\\ \require{cancel}\cancel{x^2}+\delta y&=\cancel{x^2}+2x\delta x+\delta x^2\\ \delta y&=2x\delta x+\delta x^2\\ \frac{\delta y}{\delta x}&=2x+\delta x\\ \text{Now let }&\delta x\rightarrow0\text{ leaving}\\ \frac{\delta y}{\delta x}&=2x\\ \frac{\text{d}y}{\text{d}x}&=2x \end{align}
And to me, this is just treating $\frac{\delta y}{\delta x}$ as a fraction.
I know that technically you're doing $0/0$ if you think about it, but are there any examples where treating $\text{d}y/\text{d}x$ is really inappropriate?
