Let $(A,V)$ be an affine space over a finite dimensional real vector space $V$. $A$ is homeomorphic to $V$, and thus a topological manifold. It is also easily equipped with a smooth structure to make it a smooth manifold. Also, its tangent bundle is trivial and can be identified as $A\times V$. A differentiable map $f:A\to B$ between affine spaces $(A,V),~(B,W)$ then induces a map $$\begin{align}\mathrm df:&A\times V\to B\times V,\\ &(P,v)\mapsto(f(P),\mathrm df_P(v)). \end{align}$$ This is really the trivial case of the usual differential $\mathrm df:TM\to TN$ of a function between manifolds $M,N$. But it's slightly better than the seemingly even more trivial case of the differential of a function between vector spaces because affine spaces already come equipped with their own custom made tangent spaces and associated trivial tangent bundle.
So my question is: could we model the tangent bundle on this trivial tangent bundle over an affine space, instead of constructing tangent spaces out of thin air (or maybe thick air, but the fact that we have several different standard constructions makes it seem a bit unnatural) and gluing them together to get the tangent bundle? Something along the lines of: a tangent bundle $TM$ is a vector bundle $TM\to M$ whose local trivializations are homeomorphic to $U\times V$, where $U$ is an open subset of an affine space over $V$, and whose coordinate changes are induced by the differential of the coordinate changes of $M$, that is, if $\phi$ is a coordinate change of $M$, then $(\phi,\mathrm d\phi)$ is a coordinate change of $TM$.