The implicit function theorem says that if an equation system is given by $f(x,t)=0$ (where $x$ and $t$ are vectors, and where $f(x,t)$ is a vector of the same size as $x$) then $$Dx/Dt=-\operatorname{inverse}(Df/Dx)*Df/Dt.$$
For univariate systems this formula can be applied also to find higher order derivatives (e.g. http://www.informatik-forum.at/attachment.php?attachmentid=11423&d=1195488667). Does this work for multivariate systems as well?
From $0=f(x,t)$ you get the first derivative by solving $$ 0=f_x(x(t),t)·x'(t)+f_t(x(t),t). $$ Further differentiation of this equation gives an equation for the second derivative $$ 0=f_x(x(t),t)·x''(t)+f_{xx}(x(t),t)[x'(t),x'(t)]+2·f_{tx}(x(t),t)·x'(t)+f_{tt}(x(t),t) $$ This can be continued for higher derivatives, restricted only by the smoothness of $f$. One has to observe the order of operands since the derivatives are matrices and tensors, thus unlike the scalar case not commutative.