this post was submitted on 16 Oct 2023
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You can't multiply both sides by dx in much the same way you can't differentiate a duck. That said, even pure mathematicians sort of think of it that way as a useful shorthand.
Can't you just use infinitesimals and then actually multiply them? It never results in an invalid operation with the normal dx, only the one with the fancy d (forgive my lack of terminology knowledge)
In (d/dx)f(x), d/dx is a symbol that means the derivative of f with respect to x. It's not a division of two variables. But, the reason the symbol is useful is that you sort of can multiply the dx in some situations.
I understand that it's a symbol, not a fraction, and that the top and bottom are linked and not separable. But, you can also use an equivalent infintesimal fraction dy/dx with the actual infintesimal values dy and dx being manipulatable. If I'm wrong, you'll be able to find an example that doesn't work (without using partial derivatives-- those actually can't be cancelled).