Let $A\to B$ be a morphism of commutative rings. Let $\mathcal C$ be the category of commutative $A$-algebras augmented over $B$. Let $\mathcal M_B$ denote the category of $B$-modules. The cotangent complex can be defined like this (I will brush model-categorical considerations under the carpet). Consider the derived functor of the functor

$$s\mathcal C \to s\mathcal M_B$$

obtained levelwise from the functor $\mathcal C\to \mathcal{M}_B$ given by $R\mapsto B\otimes_R \Omega_{R|A}$. Here $\Omega$ denotes Kähler differentials.

The image of $B\in \mathcal C$ under this derived functor is the cotangent complex $\mathbb L \Omega_{A|B}$.

If $M$ is a $B$-module, one defines the André-Quillen cohomology modules of $A\to B$ with coefficients in $M$ as $$D^i(B|A,M)=H^q(\mathcal M_B(\mathbb L\Omega_{B|A},M)) \cong H^q \mathrm{Der}_A(P,M)$$

where $P\to A$ is a projective resolution of $A$ in $s\mathcal C$.

So I guess this allows one to safely say that André-Quillen cohomology are some kind of "derived functor of the derivations".

I am wondering if it is (sometimes?) possible to **resolve the module variable**, i.e. to derive the functor $\mathrm{Der}_A(B,-):s\mathcal M_B \to s\mathcal M_B$, and get the same result. This feels somehow like asking whether $\mathrm Der_A(-,-): s\mathcal C \times \mathcal sM_B\to s\mathcal M_B$ is balanced.