Let M be a full Hilbert C * -module over a C * -algebra A, and let End * A (M) be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End * A (M) is an inner derivation, and that if A is σ-unital and commutative, then innerness of derivations on "compact" operators completely decides innerness of derivations on End * A (M). If A is unital (no commutativity is assumed) such that every derivation of A is inner, then it is proved that every derivation of End * A (L n (A)) is also inner, where L n (A) denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist x 0 , y 0 ∈ M such that x 0 , y 0 = 1, we characterize the linear A-module homomorphisms on End * A (M) which behave like derivations when acting on zero products.

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