Faà di Bruno is Taylor Composition

We give an elementary, self-contained proof of the multi-variate Faa di Bruno formula as a composition theorem for Taylor Polynomials in the setting of real normed vector spaces: if $\phi:E\to F$ and $\psi:F\to G$ are $k$-fold Fréchet differentiable at the points $x \in E$, $y = \phi(x) \in F$ then the Taylor Polynomials compose in the natural way: $$ T^k_x(\psi\circ\phi) = \pi_{\leq k}(T^k_y(\psi) \circ T^k_x(\phi)). $$ From this composition principle we derive (i) the partition formula for higher Fréchet derivatives and (ii) the multi-index coefficient formula established by Constantine–Savits. While many recent papers frame Faà di Bruno via Bell polynomials, trees, Hopf algebras, jets, or coordinate combinatorics, our approach isolates the analytic core—functoriality of pointwise Taylor approximation—under minimal assumptions. We avoid making finite-dimensionality or completeness assumptions and work with a weak notion of pointwise Fréchet differentiability. The familiar combinatorial coefficients arise mechanically from two standard operations: polarization and coefficient extraction from composed Taylor polynomials. The result is a short, self-contained account that covers all relevant versions of the Formula in a very general setting. As an application we present a general higher-order Leibniz rule in both partition and multi-index form.