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I think of $\frac{d}{dx}$ as a function or operation which we apply because it instructs us to take $\lim_{h\rightarrow0}\frac{(f(a+h)-f(a))}{h} = f'(a)$; however, in differential equations or integration we just simply play around with $dx$ by moving it to the numerator. I'm confused. What exactly does it mean and what do the individual terms $ dx$ or $dy$ stand for without any denominator?

DDS
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Mr.HiggsBoson
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    Moving the $dx$ around is not rigorous at all (in fact I advise not doing so). However, it serves as an intuition of "a nudge in the $x$ direction". So in derivatives $df/dx$ "measures" how much f wiggles as I nudge $x$. In integration, it is interpreted as how much area is under a curve on an interval of a "nudge in $x$". I think most of all $dx$ is just the intuitive analog to $\Delta x$ in the discrete version of calculus (slopes and sums) to the continuous version of calculus (instantaneous slopes and areas) – D.R. Jun 28 '19 at 19:43
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    When undergrad differential equations textbooks do calculations that involve manipulating $dx$ and $dy$, you can always rephrase the calculations in a slightly different way to avoid treating $dx$ and $dy$ as separate quantities. – littleO Jun 28 '19 at 19:46
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    Where it works it is an application of the chain rule (for instance in the substitution rule of integration), which itself has this nice "cancellation" pattern in the Leibniz notation. – Lutz Lehmann Jun 28 '19 at 21:17
  • See also https://math.stackexchange.com/questions/27425/what-am-i-doing-when-i-separate-the-variables-of-a-differential-equation and the linked questions there. – Hans Lundmark Jun 29 '19 at 06:44

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