I have two smooth functions $f, \phi: \mathbb{R} \to \mathbb{R}$ such that $\phi$ is invertible with smooth inverse $\phi^{-1}$. I want to know the $n$-th derivative of the composite function $\phi^{-1} \circ f \circ \phi$.
More precisely, supposing that $f(0) = \phi(0) = \phi^{-1}(0) = 0$, I'm looking for a formula for $$(\phi^{-1} \circ f \circ \phi)^{(n)}(0)$$ in terms of just the derivatives for $f$ and $\phi$ (and $\phi^{-1}$).
For the composite of two functions, Faà di Bruno's formula works out the combinatorics of the $n$-th derivative, but applying the formula twice gives a mess that doesn't seem to simplify much in my special case. There is also an answer here, but it seems to have a gap, and again it doesn't use the fact that my composite is obtained by conjugating a function.