  In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem
to compute, in terms of the maximal eigendata of the Ruelle operator, the
pressure derivative of translation invariant spin systems taking values on a
general compact metric space. On this setting the absence of metastable states
for continuous potentials on one-dimensional one-sided lattice is proved. We
apply our results, to show that the pressure of an essentially one-dimensional
Heisenberg-type model, on the lattice $\mathbb{N}\times \mathbb{Z}$, is
Fr\'echet differentiable, on a suitable Banach space. Additionally, exponential
decay of the two-point function, for this model, is obtained for any positive
temperature.
