What does d(something) mean? In a book I am reading on differential equations, the author writes the following:

$$e^{\int P(x) \mathrm{d}x}\mathrm{d}y+P(x)e^{\int P(x) \mathrm{d}x}y\mathrm{d}x=Q(x)e^{\int P(x) \mathrm{d}x}\mathrm{d}x$$
  A shrewd observer now discerns that the left side is indeed
  $$\mathrm{d}\left(e^{\int P(x) dx}y\right)$$

What does this mean? This book was published around 1980s, so it may use different notation, but I am sure there is a meaning for this expression. What is it?
 A: $\omega=df(z)$ is the same thing as $\omega=f'(z)dz$. It is essentially saying to use this rule with respect to $x$ so that $$d(e^{\int P(x)dx}y)=(e^{\int P(x)dx}y'+yP(x)e^{\int P(x)dx} ) dx$$
And noting $y'(x)dx=dy$ by the same rule.
A: You can "divide through" by $dx$, then change the statement to "A shrewd observer now discerns that the left side is indeed $\frac{d}{dx} \left(e^{\int P(x) \, dx} y \right)$."
A: Basically $dx, dy$, etc. where $x$ and $y$ are variables refer to differential elements. You can read more in the Wiki here: http://en.wikipedia.org/wiki/Differential_%28infinitesimal%29
In this case, the observation that expression $d(e^{\int P(x)dx}y)$ is equal to the LHS is shorthand for asserting that:
$\frac{d(e^{\int P(x)dx}y)}{dx} = y \cdot P(x)\int e^{P(x)}dx + \int e^{P(x)}dx \cdot \frac{dy}{dx}$
which is a consequence of applying the Product Rule. Note that Chain Rule is also being applied, since:
$\frac{d}{dx}(e^{\int P(x) dx}) = P(x) \cdot (e^{\int P(x) dx}) $
A: Let $f = f(x_1, x_2, \ldots, x_n)$ be a function of $n$ variables. Then,
$$ \mathrm{d}f = \sum_{i = 1}^n \frac{\partial f}{\partial x_i} \mathrm{d}x_i$$
In this case, $f = f(x,y)$, so $$\mathrm{d}f = \frac{\partial f}{\partial x} \mathrm{d}x + \frac{\partial f}{\partial y} \mathrm{d}y.$$

For more information, see: http://en.wikipedia.org/wiki/Total_derivative.
