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Worse, I like to cancel out the "d"s.
If I remeber right you can do that, because there’s a longer and proper way that arrives at the same conclusion
Yes, and what is aimed for is separation of variables to find a solution of this differential equation. (The differential operator d should be printed as upright letter btw)
There’s no reason it shouldn’t work.
dy/dx is the same as (y1 - y2) / (x1 - x2) as the distance between the two points approaches zero. “dx” and “dy” aren’t very useful measurements on their own though.
Yes, yes we can!
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).
