I'm working with an infinitely differentiable function $\phi\colon\mathbb{R}^{n}\to\mathbb{R}$ and an infinitely differentiable bijection $F$ from a subset $\mathbb{R}^{n}$ to a subset of $\mathbb{R}^{n}$.
I'm trying to figure out how to write the $m$ derivative of $\phi\circ F^{-1}$ at a point $x$. Via the chain rule, I computed the first derivative as $(D\phi_{i}(F^{-1}(x))) \circ DF^{-1}(x)$. I'm do not know how to do the second derivative or write down a general formula for higher derivatives. So, how do you do it for the second and in general?
Furthermore, I know that since $DF^{-1}(x) = (DF(x))^{-1}$ that I can write down the first derivative of the function in question in terms of $D\phi$, $F^{-1}(x)$ and $DF(x)$. I would like to have a general formula that can written only in terms of these quantities too. Since I cannot write down the 2nd derivative, I've not made much progress here and since this is important to the area that I'm working in, I would like to know if this is even possible and if so how?
Thanks for any help that you can give me. If there are any good references on these types of derivatives, those would also be appreciated.