This derivation reveals a concise expression for derivatives of composite functions across multiple dimensions, extending existing formulas.
By applying Taylor expansion and the Leibniz rule for partial derivatives, we provide a straightforward deduction of the multivariate Faà di Bruno formula, yielding a concise and simple expression for the derivatives of the composite function g∘ f Rⁿ g ∘ f : R n → R , with f Rⁿᵐ f : R n → R m and g Rᵐ g : R m → R . Our formulation extends the original Faà di Bruno formula to the multidimensional case in a natural way.
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Boiti et al. (2026) studied this question.
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