Gauss' Law with Electric displacement vector

I have a doubt about the applying Gauss’ Law for Electric displacement.

From what I know, $$vec{nabla} cdot vec{D} = vec{nabla} cdot (epsilon_0vec{E} + vec{P})$$

And Gauss’ Law states that:
$$
oint_S vec{D} cdot dvec{S} = Q_f
$$

where $Q_f$ is the total free charge inside a volume.

Intuitively, this makes some sense because $epsilon_0$ is in LHS part of the equation.

What confuses me here is that my professor said that Divergence Theorem is used for this conversion, and I do not understand how its being used. Could someone please explain. It would be very helpful, thanks!!