generalizations of the Leibniz rule

For the derivative, the product rule

$${(fg)}^{\prime }=f^{\prime }g+f{g}^{\prime }$$

is known as the Leibniz rule. Below are various ways itcan be generalized.

Higher derivatives

Let $f,g$ be real (or complex)functions defined on an open interval of $ℝ$. If$f$ and $g$ are $k$ times differentiable, then

$${(fg)}^{(k)}=\sum _{r=0}^k\left(\genfrac{}{}{0pt}{}{k}{r}\right)f^{(k-r)}{g}^{(r)}.$$

Generalized Leibniz rule for more functions

Let $f_1,\mathrm{\dots },f_r$ be real (or complex) valued functionsthat are defined on an open interval of $ℝ$.If $f_1,\mathrm{\dots },f_r$ are $n$ times differentiable, then

$$\frac{d^n}{d{t}^n}\prod _{i=1}^r{f}_i(t)=\sum _{n_1+\mathrm{\cdots }+n_r=n}\left(\genfrac{}{}{0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_r}\right)\prod _{i=1}^r\frac{d^{n_i}}{d{t}^{n_i}}{f}_i(t).$$

where $\left(\genfrac{}{}{0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_r}\right)$ is the multinomial coefficient.

Leibniz rule for multi-indices

If $f,g:{ℝ}^n\to ℝ$ are smooth functions defined on anopen set of ${ℝ}^n$, and $j$ is a multi-index, then

$${\partial }^j(fg)=\sum _{i\le j}\left(\genfrac{}{}{0pt}{}{j}{i}\right){\partial }^i(f){\partial }^{j-i}(g),$$

where $i$ is a multi-index.