Why can we "separate" variables? 
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What am I doing when I separate the variables of a differential equation? 

My school textbook has a section on differential equations. One of the tricks used is the following-
$$\frac{dx}{dy}=\frac{x}{y}\implies\frac{dx}{x}=\frac{dy}{y} $$ Integration is then duly carried out.Sparation of the variables leaves an impression on me that somehow, $dy$ is "dividing" $dx$. Whereas,when I studied the definition of the derivative, it was like $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.$$
I am not convinced how the so-called separation of variables is legal .Does it follow from the definition of the derivative ? Can anyone guide me to a proof?
 A: Here is a rough explanation: It follows because the derivative is a limit of a quotient of small values that, in fact, may be separated.  That is, think of 
$$\frac{dx}{dy}=\frac{x}{y}\implies\frac{\Delta x}{x} \approx \frac{\Delta y}{y}$$
Now imagine the act of integrating both sides in the latter equation as actually being one of summing both sides over all values of $x$ and $y$ in a given interval.  This sum, as it turns out, will become a Riemann sum in the limit of extremely small steps $\Delta x$ and $\Delta y$, which of course leads to our integrals which we got from separation of variables.
A: There's a notion called a "differential form". If $x$ and $y$ are functionally dependent and differentiable, then it turns out that the differential forms $dx$ and $dy$ are (more or less) multiples of each other, and the ratio happens to be $$ dx = \frac{dx}{dy} dy$$ This fact isn't really just cancelling the numerator and denominator.
Before I learned differential forms, I imagined a term "$dz$" as being a derivative with respect to some variable I hadn't decided yet. So I had interpreted the equation above as really meaning
$$ \frac{dx}{du} = \frac{dx}{dy} \frac{dy}{du} $$
where I hadn't really decided what variable I wanted to use for $u$.
