Say we have a derived scheme over an algebraically closed field $X/k$, viewed as a functor $X : \operatorname{Aff}_k^{\operatorname{op}} \to \infty\operatorname{-Grpd}$ and we know its formal neighborhood around every geometric point $X^{\land}_x$ (e.g. $\operatorname{char}k=0$ and we know how to compute the corresponding tangent complex) as well as its restriction to discrete rings $X_0 := X\mid \operatorname{Aff}_k^{\text{cl}}$. Will $X$ be determined uniquely by this data, and if so - is there an explicit way of computing its values on a general derived affine scheme? In the situation I have in mind $X_0$ is a classical scheme (i.e. $X(\pi_0(R)) \in \operatorname{Sets}$ and does not have higher homotopy groups).