I'm just starting my DE class, although I've been familiar with separable DEs for a while. Although they're (so far) pretty straight-forward to solve, I don't really understand the theory behind seperable DEs. In calc 1, it was stressed that dy/dx is NOT a fraction that can be "taken apart." Looking at the definition of the derivative, it's clear that you cannot rewrite the limit as one limit divided by another limit, because the denominator would be 0, breaking a limit law. It seems to me that the point of derivatives is that we have this indeterminate 0/0 form that, given the context of the original function, we can solve. It seems to me that separating the limit would be like saying that the dy doesn't depend on the dx. This seems like an "abuse of notation" to me, just like the one often used to "proof" the derivative chain rule. Can somebody please help me clear this up?