We all know that since $\textrm{d}y$ and $\textrm{d}x$ are not numbers, we can't treat $\frac{dy}{dx}$ as a ratio. However, in various operations, this treatment seems to work. I've picked a few examples:

* **Chain Rule:**
$$\frac{du}{dx} = \frac{du}{dy} \frac{dy}{dx}$$
* **Variable separation:**
$$ \frac{dy}{dx} = f(x)h(y)$$
$$\frac{dy}{h(y)} = f(x)dx$$
* **Integral of $\frac{dy}{dx}$**:
$$ \int \frac{dy}{dx}dx = \int dy = y$$
And there is also some examples that this treatment does not work:
$$\frac{dy}{dx} = - \frac{\partial F / \partial y}{\partial F / \partial x}$$
Where the minus sign shows that this is a counterexample. Now, if this isn't correct, why does it works for most cases? And, if it works for most cases, why there are a few cases that it won't work? Thanks.

**References**

1) Answer from asmeurer in [Is $\frac{dy}{dx}$ not a ratio?](https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio/21340#21340)