  Following Feigin and Fuchs, we compute the first cohomology of the Lie
superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional
real superspace with coefficients in the superspace of linear differential
operators acting on the superspaces of weighted densities. We also compute the
same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give
1-cocycles spanning these cohomology. We classify generic formal
$\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module
structure on the superspaces of symbols of differential operators. We prove
that any generic formal $\mathfrak{osp}(1|2)$-trivial deformation of this
$\mathcal{K}(1)$-module is equivalent to a polynomial one of degree $\leq4$.
This work is the simplest superization of a result by Bouarroudj [On
$\mathfrak{sl}$(2)-relative cohomology of the Lie algebra of vector fields and
differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127].
Further superizations correspond to $\mathfrak{osp}(N|2)$-relative cohomology
of the Lie superalgebras of contact vector fields on $1|N$-dimensional
superspace.
