  We study the differential symmetric signature, an invariant of rings of
finite type over a field, introduced in a previous work by the authors in an
attempt to find a characteristic-free analogue of the F-signature. We compute
the differential symmetric signature for invariant rings $k[x_1,\dots,x_n]^G$
where $G$ is a finite small subgroup of $\mathrm{Gl}(n,k)$ and for hypersurface
rings $k[x_1,\dots,x_n]/(f)$ of dimension $\geq3$ with an isolated singularity.
In the first case, we obtain the value $1/|G|$, which coincides with the
F-signature and generalizes a previous result of the authors for the
two-dimensional case. In the second case, following an argument by Bruns, we
obtain the value $0$, providing an example of a ring where differential
symmetric signature and F-signature are different.
