Cadabra manual: Product

product_rule

Apply the Leibnitz rule to a derivative of a product

Apply the product rule or "Leibnitz identity" to an object which has the Derivative property, i.e.

D{#}::Derivative; ex:=D(f g); product_rule(_);

\(\displaystyle{}\text{Attached property Derivative to }D{\#}.\)

\(\displaystyle{}D\left(f g\right)\)

\(\displaystyle{}D{f} g+f D{g}\)

This of course also works for derivatives which explicitly mention indices or components, as well as for multiple derivatives, as in the example below.

D{#}::Derivative. ex:=D_{m n}(f g);

\(\displaystyle{}D_{m n}\left(f g\right)\)

product_rule(_);

\(\displaystyle{}D_{m}\left(D_{n}{f} g+f D_{n}{g}\right)\)

distribute(_);

\(\displaystyle{}D_{m}\left(D_{n}{f} g\right)+D_{m}\left(f D_{n}{g}\right)\)

product_rule(_);

\(\displaystyle{}D_{m}{D_{n}{f}} g+D_{n}{f} D_{m}{g}+D_{m}{f} D_{n}{g}+f D_{m}{D_{n}{g}}\)