  Let $A$ be a finite-dimensional algebra over a field $F$ with char$(F)\ne 2$.
We show that a linear map $D:A\to A$ satisfying $xD(x)x\in [A,A]$ for every
$x\in A$ 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$(F)\ne 3$, we show that a linear map $T:A\to A$ satisfies $T(x)^3- x^3 \in
[A,A]$ for every $x\in A$ if and only if there exist a Jordan automorphism $J$
of $A$ lying in the multiplication algebra of $A$ and a central element
$\alpha$ satisfying $\alpha^3=1$ such that $T(x)=\alpha J(x)$ for all $x\in A$.
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$(F)\ne 2,3$) is a Jordan automorphism.
