Comparison of prismatic cohomology and derived de Rham cohomology

We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over ${\mathcal{O}}_K$ with $K$ being a $p$-adic field, we improve Breuil–Caruso’s theory on comparison between torsion crystalline cohomology and torsion 'etale cohomology.