  The multidimensional chain rule formula for analytic functions and its
generalisation to higher derivatives perfectly work in the algebraic setting in
characteristic zero. In positive characteristic one runs into problems due to
denominators in these formulas. In this article we show a direct analog of
these formulas using higher derivations which are defined in any
characteristic. We also use these formulas to show how higher derivations to
different coordinate systems are related to each other. Finally, we apply this
to polynomial automorphisms in arbitrary characteristic and obtain a formula
for the inverse of such a polynomial automorphism.
