Suppose that I have some map $F\colon K\to\mathbb{R}^{n}$ where $K\subset\mathbb{R}^{n}$ and $K$ is compact. Let $\alpha = (\alpha_{1},\cdots,\alpha_{n})$ be some multi-index. Suppose that $F$ is $C^{|\alpha|}$ and has an inverse in $C^{|\alpha|}$. Further suppose that for a fixed point $x_{0}$, we know $F^{-1}(x_{0})$. Further, suppose we also know all the derivatives of $F$. What is the formula for $(\frac{\partial}{\partial x^{\alpha}})F^{-1}(x_{0})$ in terms of this information.
I did find something that might help via a previous question: http://www.sciencedirect.com/science/article/pii/S0022247X86800099 but I was unable to extend the results therein.
The following might also be helpful:http://www.ams.org/journals/tran/1996-348-02/S0002-9947-96-01501-2/S0002-9947-96-01501-2.pdf