Let $A \in M_n (\mathbb Z)$ such that $A^{-1} \in M_n(\mathbb Z)$. The torus $\mathbb T^n$ is $\mathbb R^n/\sim$ where $x \sim y \iff x-y\in \mathbb Z^n$. $A$ defines a diffeomorphism $f: \mathbb T^n \to \mathbb T^n$ by $f([x]_\sim) = [Ax]_\sim$. What is the differential $D_x f$ of $f$?
Using the definition of the tangent space as equivalence classes of curves I tried calculating $D_x f ([\gamma_i])$ where $\gamma_i (t) = x + te_i$ form a basis for $T_x \mathbb T^n$, but couldn't get to something meaningful.