Let $S$ be a base scheme. For which algebraic stacks $X$ over $S$ can we define a sheaf of differentials $\Omega^1_{X/S}$ (classifying derivations)? Probably it works when $X$ is Deligne Mumford because $\Omega^1$ is stable under pullbacks of étale morphisms? But for example for the classifying stack $B \mathbb{G}_m$ there is no $\Omega^1$ because deformations of invertible sheaves have non-trivial automorphisms, right?

Even if there is no $\Omega^1_{X/S}$ in general, I would like to know if there is a "tangent bundle" $T(X/S)$ which satisfies the adjunction $\hom_S(Y[\varepsilon]/\varepsilon^2,X) \simeq \hom_S(Y,T(X/S))$ for algebraic stacks $Y$ over $S$. If $\Omega^1_{X/S}$ exists then one may take $T(X/S) = \mathrm{Spec} \mathrm{Sym} \Omega^1_{X/S}$, but perhaps this definition is "too discrete" for algebraic stacks. Perhaps one can encode the cotangent complex into $T(X/S)$?

Edit: At this moment I am not interested in derived stacks. Algebraic stacks mean Artin stacks in the usual sense.