  The derivation constant $K(A)\geq \frac{1}{2}$ has been extensively studied
for \emph{unital} non-commutative $C^*$-algebras. In this paper, we investigate
properties of $K(M(A))$ where $M(A)$ is the multiplier algebra of a non-unital
$C^*$-algebra $A$. A number of general results are obtained which are then
applied to the group $C^*$-algebras $A=C^*(G_N)$ where $G_N$ is the motion
group $\R^N\rtimes SO(N)$. Utilising the rich topological structure of the
unitary dual $\widehat{G_N}$, it is shown that, for $N\geq3$, $$K(M(C^*(G_N)))=
\frac{1}{2}\left\lceil \frac{N}{2}\right\rceil.$$
