Let A be a finite-dimensional algebra over a field F with char(F)≠2. We show that a linear map D:A→A satisfying xD(x)x∈A,A for every x∈A is the sum of an inner derivation and a linear map whose image lies in the radical of A. Assuming additionally that A is semisimple and char(F)≠3, we show that a linear map T:A→A satisfies T(x)3−x3∈A,A for every x∈A if and only if there exist a Jordan automorphism J of A lying in the multiplication algebra of A and a central element α satisfying α3=1 such that T(x)=αJ(x) for all x∈A. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra A (over a field F with char(F)≠2,3) is a Jordan automorphism.