> [!theorem] > > Let $X, Y$ and $Z$ be $C^p$-[[Manifold|manifolds]] modelled on the [[Banach Space|Banach spaces]] $E, F$, and $G$ respectively. Let $f: X \to Y$ and $g: Y \to Z$ be $C^p$ [[Manifold Morphism|morphisms]], and $p \in X$, then the [[Differential|differential]] inherits the [[Chain Rule|chain rule]] from ordinary derivatives: > $ > d(f \circ g)_p = dg_{f(p)} \circ df_p > $ > *Proof*. Let $(U, \psi)$, $(V, \varphi)$, and $(W, \xi)$ be [[Atlas|charts]] containing $p$, $f(p)$, $(f \circ g)(p)$, respectively, then $f_{U, V}$ and $g_{V, W}$ are $C^p$. By the chain rule, > $ > D\braks{(f \circ g)_{U, W}}(\hat p) = D(g_{V, W})(\widehat{f(p)}) \circ D(f_{U, V})(\hat p) > $ > where for any $h \in T_{(U, \psi, p)}X$, > $ > \begin{align*} > dg_{f(p)}\circ df_p (\ol{h}) &= dg_{f(p)}\braks{\ol{D(f_{U, V})(\hat p)(h)}} \\ > &= \ol{\braks{D(g_{V, W})(\widehat{f(p)}) \circ D(f_{U, V})(\hat p)} \cdot (h)} \\ > &= \ol{\braks{D\braks{(f \circ g)_{U, W}}(\hat p)}(h)} \\ > &= d(f \circ g)_{p}(\ol{h}) > \end{align*} > $