  In this paper we describe the tangent vectors of the stable and unstable
manifold of a class of Anosov diffeomorphisms on the torus $\mathbb{T}^2$ using
the method of formal series and derivative trees. We start with linear
automorphism that is hyperbolic and whose eigenvectors are orthogonal. Then we
study the perturbation of such maps by trigonometric polynomial. It is known
that there exist a (continuous) map $H$ which acts as a change of coordinate
between the perturbed and unperturbed system, but such a map is in general, not
differentiable. By "re-scaling" the parametrization $H$, we will be able to
obtain the explicit formula for the tangent vectors of these maps.
