The paper consists of four
sections, the first of which is an introduction to the problems and a survey of the
results known. The second section develops and supplies some new proofs of the
fundamental classic formulas deriving another explicit formula for the n-th derivative
of composite functions. The third section derives new explicit formulas for the n-th
Lie derivative, i. e., for the n-th derivative of composite functions, defined implicitly
by the parametric representation w = g(t), z = f(t) and, in particular, for
the n-th derivative of inverse functions. Compared to the classic formula of
Lagrange, the Taylor coefficients of the parametrically given composite functions
are here determined by new formulas as explicit functions of the Taylor
coefficients of the two component functions. In particular, the respective
explicit inverses in the famous class S of regular schlicht functions in the
unit disk are found. Moreover, an explicit expression for the substitution
of the higher derivatives in Legendre transformations has been given. The
fourth section points out the conditions under which all result proved in the
previous sections remain valid and are in the real domain. Also, it is noted
that the corresponding results remain valid and are for the formal power
series.
