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The derivative of a function $y = f(x)$, $\frac{dy}{dx}$ seems to behave like a quotient in many cases:

$$ dy = \frac{dy}{dx} dx,$$

or

$$ u = h(x) $$ $$\int f(x) dx = \int h(f(x)) \frac{du}{dx} du$$

Yet we're often told that it's not correct to view it that way. For example, the wikipedia article on Leibniz' notation says

The expression dy/dx should not be read as the division of two quantities dx and dy.

Is there an example of a situation in which viewing $\frac{dy}{dx}$ as a quotient will lead to an incorrect result?

jforberg
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    It's not that viewing it that way gets you an incorrect result (AFAIK), it's just that $dy/dx$ actually isn't a quotient of two quantities, which is why we're encouraged not to think of it as such. –  Sep 13 '16 at 21:44
  • Treating partial derivatives as quotients will lead to mistakes. Without a lot more work than is arguably necessary, what $dy$ and $dx$ are is not even defined. It can be treated as a fraction in some sense, but the naive approach tends to push too much under the rug, in terms of the chain rule as well as implicit differentiation. In particular, we need to distinguish between functions and variables, which the Leibniz notation is not well suited for. – Chill2Macht Sep 13 '16 at 21:47
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    Also of interest: http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio – Simply Beautiful Art Sep 13 '16 at 21:57
  • @GFauxPas and Simple art, thanks for the links which explained it very well. Sorry for the repost. – jforberg Sep 13 '16 at 21:58

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