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If you work out deriving the quotient rule yourself using the exact trick you're highlighting, you can see that the quotient rule is nothing more than the product rule and the chain rule used together. If you rearrange a quotient to use the product rule, then you'll very likely be using the chain rule shortly thereafter on the $(\cdots)^{-1}$ part, and you will inevitably arrive at exactly the same result as if you had used the quotient rule but with more steps. The only potential difference is whether your result looks like $\frac{a}{b} + \frac{c}{b^2}$ or $\frac{ab + c}{b^2}$—but of course, as I'm sure you can see, that's no difference at all.