  We determine the $\sigma$-derivations of quantum tori and quantum affine
spaces for a toric automorphism $\sigma$. By standard results, every toric
automorphism $\sigma$ of a quantum affine space $\mathcal{A}$ and every
$\sigma$-derivation of $\mathcal{A}$ extend uniquely to the corresponding
quantum torus $\mathcal{T}$. We shall see that, for a toric automorphism
$\sigma$, every $\sigma$-derivation of $\mathcal{T}$ is a unique sum of an
inner $\sigma$-derivation and a $\sigma$-derivation that is conjugate to a
derivation and that the latter is non-zero only if $\sigma$ is an inner
automorphism of $\mathcal{T}$. This is applied to determine the
$\sigma$-derivations of $\mathcal{A}$ for a toric automorphism $\sigma$,
generalizing results of Alev and Chamarie for the derivations of quantum affine
spaces and of Almulhem and Brzezi\'{n}ski for $\sigma$-derivations of the
quantum plane.
  We apply the results to iterated Ore extensions $A$ of the base field for
which all the defining endomorphisms are automorphisms and each of the adjoined
indeterminates is an eigenvector for all the subsequent defining automorphisms.
We present an algorithm which, in characteristic zero, will, for such an
algebra $A$, either construct a quantum torus between $A$ and its quotient
division algebra or show that no such quantum torus exists.
  Also included is a general section on skew derivations which become inner on
localization at the powers of a normal element which is an eigenvector for the
relevant automorphism. This section explores a connection between such skew
derivations and normalizing sequences of length two. This connection is
illustrated by known examples of skew derivations and by the construction of a
family of skew derivations for the parametric family of subalgebras of the Weyl
algebra that has been studied in three papers by Benkart, Lopes and Ondrus.
