State the inverse function theorem for a function $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$. Suppose $F$ is a differentiable bijection with $F^{-1}$ also differentiable. Show that the derivative of $F$ at any point in $\mathbb{R}^{n}$ is a linear isomorphism.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function such that the partial derivatives $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$ exist and are continuous. Assume there is a point $(a, b) \in \mathbb{R}^{2}$ for which $f(a, b)=0$ and $\frac{\partial f}{\partial x}(a, b) \neq 0$. Prove that there exist open sets $U \subset \mathbb{R}^{2}$ and $W \subset \mathbb{R}$ containing $(a, b)$ and $b$, respectively, such that for every $y \in W$ there exists a unique $x$ such that $(x, y) \in U$ and $f(x, y)=0$. Moreover, if we define $g: W \rightarrow \mathbb{R}$ by $g(y)=x$, prove that $g$ is differentiable with continuous derivative. Find the derivative of $g$ at $b$ in terms of $\frac{\partial f}{\partial x}(a, b)$ and $\frac{\partial f}{\partial y}(a, b)$.