  The main object of study of this paper is the notion of 3-Lie superalgebras
with superderivations. We consider a representation $(\Phi,\mathcal{P})$ of a
$3$-Lie superalgebra $\mathcal{Q}$ on $\mathcal{P}$ and construct first-order
cohomologies by using superderivations of $\mathcal{P},\mathcal{Q}$ which
induces a Lie superalgebra $\mathcal{T}_{\Phi}$ and its representation $\Psi$.
Then we consider abelian extensions of $3$-Lie superalgebras of the form
$0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow
\mathcal{Q}\rightarrow 0$ with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$ and
construct an obstruction class to extensibility of a compatible pair of
superderivation. Moreover we prove that a pair of superderivation is extensible
if and only if its obstruction class is trivial under some suitable conditions.
