> [!definitionb] Definition > > Let $X$ and $Y$ be $C^p$-[[Manifold|manifolds]] ($p \ge 1$) modelled on the [[Banach Space|Banach spaces]] $E$ and $F$, respectively. > > Let $f: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]], $p \in X$, and $(U, \psi) \in X$, $(V, \varphi) \in Y$ such that $\varphi \circ f \circ \psi^{-1}$ is of class [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions). The map between the [[Concrete Tangent Space|concrete tangent spaces]] > $ > Df_{U, V}(\hat p): T_{(U, \psi, p)}X \to T_{(V, \varphi, f(p))}X > $ > where > $ > h \mapsto D(\varphi \circ f \circ \psi^{-1})(\hat p)(h) > $ > is the [[Derivative|derivative]] of $f$ **interpreted with respect** to the pair of [[Atlas|charts]] $(U, \psi)$ and $(V, \varphi)$. > > $ > \begin{CD} > T_pX @>>> T_{(U, \psi, p)}X\\ > @V{df(p)}VV @VV{Df_{U, V}(\hat p)}V \\ > T_{f(p)}Y @>>> T_{(V, \varphi, f(p))}X > \end{CD} > $ > > To connect the different concrete derivatives obtained from different pairs of charts such that the above diagram commutes, let[^1] > $ > df(p): T_pX \to T_pY > $ > be a map between the [[Tangent Space|tangent spaces]], with > $ > \ol{h} \mapsto \ol{Df_{U, V}(\hat{p})(h)} > $ > where the bar refers to the corresponding [[Equivalence Class|equivalence classes]], then $df(p)$ is the **differential** of $f$ at $p$. > [!theorem] > > The differential inherits the following properties from the [[Derivative|derivative]]: > - Linear and continuous. > - [[Chain Rule (Differential)|Chain Rule]]: $d(f \circ g)_p = dg_{f(p)} \circ df_p$ > - $d(\text{Id})_p = \text{Id}$ > - If $f$ is a [[Diffeomorphism|diffeomorphism]], then $df_p$ is a [[Space of Toplinear Isomorphisms|toplinear isomorphism]] with $(df_p)^{-1} = d(f^{-1})_{f(p)}$. [^1]: See [[Well-Definedness of the Differential]]