<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g\circ f \colon \mathbb{R}^n\to\mathbb{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo lspace="0pt">:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f \colon \mathbb{R}^n\to\mathbb{R}^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo lspace="0pt">:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g \colon \mathbb{R}^m\to\mathbb{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo lspace="0pt">:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>m</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. Our formulation extends the original Faà di Bruno formula to the multidimensional case in a natural way.</p>

A direct and elementary derivation of the multivariate Faà di Bruno formula

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2026年6月19日

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C. Boiti,
R. Manfrin

摘要

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 $g\circ f \colon \mathbb{R}^n\to\mathbb{R}$ $g \circ f : R^{n} \to R$ , with $f \colon \mathbb{R}^n\to\mathbb{R}^m$ $f : R^{n} \to R^{m}$ and $g \colon \mathbb{R}^m\to\mathbb{R}$ $g : R^{m} \to R$ . Our formulation extends the original Faà di Bruno formula to the multidimensional case in a natural way.