Abstract:
The focus of the article is the classical Faa Di Bruno formula for computing higher-order derivatives of a complex function $F(u(x))$. Here is a version of the proof of this formula. Then we prove a generalization of the Faa Di Bruno formula to the case of a complex function with an inner function $u(x,y)$ depending on two independent variables. The paper presents a formula for the $n$-th derivative of a complex function, when the argument of the outer function is a vector with an arbitrary number of components (functions of one variable). The article also considers examples of finding higher-order derivatives, illustrating both the classical Faa Di Bruno formula and its generalizations.
Keywords:Faa Di Bruno's formula, $n$-th derivative of complex functions of several variables, generalizations of Faa Di Bruno's formula for these functions, Newton's binomial and polynomial formulas.
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