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For a commutative C*-algebra $\mathcal A$ with unit $e$ and aHilbert~$\mathcal A$-module $\mathcal M$, denote by End$_{\mathcal A}(\mathcalM)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$,and by End$^*_{\mathcal A}(\mathcal M)$ the algebra of all adjointable mappingson $\mathcal M$. We prove that if $\mathcal M$ is full, then each derivation onEnd$_{\mathcal A}(\mathcal M)$ is $\mathcal A$-linear, continuous, and inner,and each 2-local derivation on End$_{\mathcal A}(\mathcal M)$ orEnd$^{*}_{\mathcal A}(\mathcal M)$ is a derivation. If there exist $x_0$ in$\mathcal M$ and $f_0$ in $\mathcal M^{'}$, such that $f_0(x_0)=e$, where$\mathcal M^{'}$ denotes the set of all bounded $\mathcal A$-linear mappingsfrom $\mathcal M$ to $\mathcal A$, then each $\mathcal A$-linear localderivation on End$_{\mathcal A}(\mathcal M)$ is a derivation.

Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules