Multi-index Leibniz rule

Statement

For smooth scalar functions $f,g$ and a multi-index $\alpha$ , the Leibniz rule is

\partial^\alpha(fg)=\sum_{\beta\le\alpha} \binom{\alpha}{\beta}(\partial^\beta f)(\partial^{\alpha-\beta}g), \qquad \binom{\alpha}{\beta}=\prod_{i=1}^n\binom{\alpha_i}{\beta_i}.

The inequality $\beta\le\alpha$ is componentwise. It suffices to assume that the functions have continuous derivatives through order $|\alpha|$ .

Derivation and estimates

Repeated application of the first-derivative product rule gives the formula. At each step, the two possibilities for which factor is differentiated combine by Pascal's binomial identity. Taking absolute values yields a finite sum of derivative products, with constants depending only on the derivative order and dimension.

The same identity holds for a fixed bilinear product of finite-dimensional vector or matrix values, with the order of the two factors preserved.