Cannot Differentiate ydx?

Hi,

In class today, we worked out the following question. However, I don't understand why the teacher said we can't differentiate ydx because it will be equal to zero, as well as where the 'd' disappeared to (please see my comments in red below)

Question: Solve for the following differential equation for the original equation: $\displaystyle \frac{dy}{dx} = \frac{-y}{4+x}$

$\displaystyle \frac{dy}{dx} = \frac{-y}{4+x}$

$\displaystyle \int (4 + x)dy = \int -ydx$

$\displaystyle \int 4dy + xdy = \int -ydx$ (you cannot differentiate ydx, ydx = 0)

$\displaystyle \int 4y + xy = 0 + C$ (where did the d go? Why not $\displaystyle 4xdy + x^2dy$)?

$\displaystyle 4y + xy + C = 0$