I've done some epsilon proofs in calculus, but in the Leibniz notation dy/dx why can we treat these as variables as when solving differential equations?
The question about dx and dy notation is an old well-worn subject.
The "Nonstandard Analysis" theory leads to a strict answer. To French readers, "Une querelle des Anciens er des Modernes" : JJacquelin's Documents | Scribd
dy and dx themselves are difficult to conceive of as variables. But as a whole just means the derivative at a particular location, so it can be manipulated without requiring too much special care (until you try to separate the fraction).
The derivative, dy/dx, is not defined as a fraction but it is defined as the limit of a fraction: . One can show that "fraction properties" can be used by going back before the limit, applying the fraction property, then taking the limit. To make that a little more formal, Calculus texts then introduce the "differential", dx, separately from the derivative, then defining, dy, by "dy= (dy/dx) dx". ("Nonstandard Analysis" avoids that whole question but requires some very deep concepts in logic, introducing a completely different number system.)
f’(x) has a standard definition not involving dy and dx.
dy=f’(x)dx is a defined relation between the variables dy and dx.
For example, if f’(x)=5, dy = 5dx by definition. If dx is 10, dy=50. If dy is 10, dx=2.
dy and dx can be interpreted as increments on the tangent line.
in differential equations when solving an initial value problem for c, you have this proof in Braun where he changes f(t) to f(r), g(t) to g(s)- I'm not sure what the reasoning is, maybe you can use any arbitrary variable since when you integrate the limits will give you dependence on t an to? the time dependence of y on t also remains? and then it's assumed you can integrate both sides