I'm learning integration using substitution and the symbols $\frac{dy}{dx}$ are used as variables which is confusing me as I thought they weren't normal variables. So if I'm integrating something and have to substitute $x^2$ so that $u = x^2$ then $\frac{du}{dx} = 2x$ and now I'm at the point that's confusing to me, I multiply both sides of $\frac{du}{dx} = 2x$ with $dx$ so I get $du = 2x dx$ and finish the substitution. So is $dx$ just a normal variable here?
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1You can use those symbols as variables, but you cannot add or subtract them from other variables. For example, $dx+dy$ is legit, $dx+3$ isn't. – Kaster Oct 30 '13 at 06:25
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This is one of those frustrating things about the notation used in introductory calculus. Technically $dy$ and $dx$ are not variables, but thanks to some weird property, you can use them just as if they were variables.
See this answer for more detail.
EDIT: I should note that while you can treat $dy$ and $dx$ just as if they are variables, it is extremely important that you do NOT think of them like $dy=d\cdot y$. Doing that would land you in a world of trouble.
