The notation df denotes differential of function f. The differential df is a map
\begin{equation} df:\mathbb{R}\rightarrow \Omega^1(\mathbb{R}) \end{equation}
where Ω1(ℝ) is the set of linear maps from ℝ to ℝ. The linear map corresponding to point p∈ℝ is often written as dfp
\begin{equation} df(p)=df_p \end{equation}
Note that in less formal settings p is often omitted altogether.
You are probably used to thinking about differentiation in a somewhat different way: given a function f from ℝ to ℝ, you create a different function f' also from ℝ to ℝ. This way for every point p on the real line ℝ you get a number f'(p). In order to introduce infinitesimal quantities more formally one needs to realize that a number, like f'(p) can be regarded as a linear map from ℝ to ℝ. This is more abstract, but what we're doing is identifying real number a with the linear map from ℝ to ℝ which takes its sole argument and returns its product with a. The simplest case to consider is the identity function which maps x to x. 586
It's derivative is equal to 1 for every point on the real line. Therefore, dxp is equal to an identity transformation for every point p on the real line.
Now, in order to define dfp explicitly for an arbitrary function f, we notice that the set of all linear maps from ℝ to ℝ is a one-dimensional vector space. Any non-zero linear map from ℝ to ℝ can be chosen to be its sole base vector, but the choice of dxp, i.e. the differential of the identity function, is most convenient. The differential of an arbitrary differentiable function f is then defined as
\begin{equation} df_p=f'(p)dx_p \end{equation}
This equation can be used to prove that d operator obeys similar laws to those for ordinary derivative. In particular,
\begin{equation} d(f \cdot g)_p = (f \cdot g)'dx_p = f'(p)\cdot g(p) dx_p + f(p)\cdot g'(p)dx_p = g\cdot df_p + f\cdot dg_p \end{equation}
Similarly, for constant c one obtains
\begin{equation} d(c\cdot f)_p = c \cdot df_p \end{equation}
If you apply these to the gas equation, you're going to obtain exactly the same result as in your book.
The purpose of differentials is to provide a formal way to think about infinitesimal quantities. For many practical purposes your intuitive reasoning and "division by dt" is often sufficient.
The usefulness of this construction becomes more apparent for the general n-dimensional case.
For more details see wikipedia articles on differential, differential forms and exterior algebra. 418