This article investigates derivation theory within the framework of hoop algebras. We introduce and examine f-derivations and (f, g)-derivations, exploring their properties and providing illustrative examples. We demonstrate that, under specific conditions, the sets of these derivations form bounded distributive lattices and bounded ∨-semilattices. Furthermore, we investigate s-derivations, where s is a square root on the hoop algebra. Notably, we prove that under certain conditions, the set of fixed points of the s-derivation coincides with the set of idempotents of the hoop algebra. This work contributes to the understanding of derivation structures in hoop algebras, particularly focusing on the interplay between derivations, square roots, and idempotent elements.