Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of time $t$, $x = x(t)$. By the multivariate chain rule,
$$
\partial_x f \cdot x'(t) + \partial_t f = 0 \;.
$$
This yields an explicit formula for $x'(t)$, if $\partial_x f \neq 0$ at the point of evaluation.
A similar argument works also in the multivariate case, when $\mathbf{x} \in \mathbb{R}^n$. For any time $t$, think of $F(\cdot, t)$ as an operator on $\mathbb{R}^n$, and the argument follows.
I'm interested in derivatives $\frac{d^l\mathbf{x}}{dt^l}$ of higher orders $l > 0$, in the multivariate case. Based on the above intuition, one might expect a similar formula to exist, allowing to calculate $\frac{d^l\mathbf{x}}{dt^l}$ in terms of the derivative tensors of $F$. The purpose being, to calculate high-order time derivatives at a given operator root.
Question:
Before setting off to derive such a formula (if exists), is there such a result in the literature? Perhaps something similar?
Edit:
I am aware of the Faà di Bruno's formula, and of its multivariate counterparts.
Edit 2:
While an expression for the first-order derivative $\frac{d\mathbf{x}}{dt}$ is commonly found in the literature when discussing the implicit function theorem as @RyanBudney notes, this question is about derivatives of higher orders.
Best Answer
Although this question sounds quite innocent, a systematic treatment of higher order derivatives of implicit functions is quite involved. On a second thought, this is no surprise if you think about how complicated higher derivatives of concatenations $f\circ g$ get (see Faà di Bruno's formula). For higher order derivatives of implicit functions, you can have a look at The Combinatorics of Higher Derivatives of Implicit Functions.