I was always taught that $\frac d {dx} (\ln x) = \frac 1 x$. No derivative had as a result any $dx$ words.
In a physics book I encountered something like this (error discussion) [there might be a little difference, as I don't have the book right now]: 

We have $\ln a + \ln b = \ln c$. So after deriving both sides we get $\frac {da} a + \frac {db}b = \frac {dc}c $

Why is this small difference $da, db, dc$ put in there? Why isn't it $\frac 1a + \frac 1b = \frac 1c $ ? And what are we differentiating in reference to? $x$, $a$ or what? 

Another example [this is exactly like this]: We have 
$[(\frac 1 {S\_1}+\frac 1 {S\_2})\rho g V + p\_0]V = nRT$. Let's treat V as a function of T and differentiate both sides wrt T. Then we have

$2(\frac 1 {S\_1}+\frac 1 {S\_2})\rho g V \frac {dV} {dT} + p\_0 \frac {dV} {dT} = nR$

Once again, why are we adding this difference quotient $\frac {dV} {dT}$? Is this as the derivative of the internal function?