----- > [!proposition] Proposition. ([[derivative of inverse function is inverse matrix]]) > Let $\Omega \subset \mathbb{R}^{n}$ be [[open set|open in]] $\mathbb{R}^{n}$. Suppose $f:\Omega \to \mathbb{R}^{m}$ is [[derivative|differentiable]] and [[inverse map|invertible]] with [[derivative|differentiable inverse]]. Then for all $p \in \Omega$, $D[f^{-1}](p)=[Df]^{-1}\big(f(p) \big).$ > [!proof]- Proof. ([[derivative of inverse function is inverse matrix]]) > Set $g:=f^{-1}$ and $y:=f(p)$. Then $f \circ g = \text{id}$. Differentiate both sides to obtain $D[g \circ f] = \b I $ and applying the [[chain rule]] we get $D[g \circ f]=Dg |_{y} Df |_{p} = D[f^{-1}] Df = I$ and now right-multiply both sides by $[Df]^{-1}$ to obtain the result. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```