  We consider the algebra $A_N=k\langle x, y\rangle/(yx-xy-x^N)$, with $k$ a
field of characteristic zero and $N$ a positive integer. Our main result is a
complete description of the first Hochschild cohomology
$\operatorname{HH}^1(A_N)$ of $A_N$ that consists both of explicit derivations
of $A_N$ whose cohomology classes span it and a description of its Lie algebra
structure. As we do this, we compute the automorphism group of the algebra, as
well as certain characteristic subgroups thereof related to locally nilpotent
derivations, classify the finite groups that act on $A_N$ and, finally, show
that there are no inner-faithful actions of generalized Taft Hopf algebras on
$A_N$. We establish this last result thanks to another calculation of
Hochschild cohomology, now with twisted coefficients.
