Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We could consider homotopic morphisms and get the homotopy category [dg-mod-$A$]. Moreover we could invert quasi-isomorphisms in [dg-mod-$A$] and obtain its derived category. Let us denote this category by $D(A)$.

On the other hand, we could consider the dg-category DG-MOD-$A$: its objects are right $A$-dg-modules and its morphisms are chain complexes of maps between dg-modules.

We notice that there is a general construction of the derived category of a dg-category $\mathcal{C}$: We first consider the dg-category of right modules over $\mathcal{C}$, which is the dg-category of contravariant dg-functors from $\mathcal{C}$ to Ch$(k)$. Let us denote this dg-category by DGM-$\mathcal{C}$. We also consider the full dg-subcategory Acycl-$\mathcal{C}$, which consists of dg-functors $\mathcal{F}:\mathcal{C}\to \text{Ch}(k)$ such that $\mathcal{F}(c)$ is acyclic for any object $c\in \mathcal{C}$. Then the derived category of $\mathcal{C}$ is given by the Verdier quotient $$ [\text{DGM}-\mathcal{C}]/[\text{Acycl}-\mathcal{C}] $$ where $[\cdot]$ is the homotopy category of dg-categories. We denote the derived category of $\mathcal{C}$ by $D(\mathcal{C})$. We could show that the Yoneda functor induces a fully faithful functor $[\mathcal{C}]\to D(\mathcal{C})$.

Now go back to the dg-algebra $A$ and the dg-category DG-MOD-$A$.

My question is: is the derived category $D(A)$ equivalent to the derived category $D(\text{DG-MOD}-A)$? Why?