Solution
= Solution
The <inverse function theorem> states that if $f:U\to\mathbb R^n$ is $C^1$ on an <open set> and $Df(a)$ is invertible, then there are <open neighbourhoods> $U_0$ of $a$ and $V_0$ of $f(a)$ such that $f:U_0\to V_0$ is a $C^1$ <diffeomorphism>. Its inverse satisfies
$$
\boxed{D(f^{-1})(f(x))=(Df(x))^{-1},\qquad x\in U_0.}
$$
If $f$ is $C^k$ or <smooth>, the inverse has the same regularity.