What is the gradient of a product of functions? The ordinary product rule for differentials is \begin{equation} d(fg) = g\,df + f\,dg \end{equation} and under the correspondence \begin{equation} df = \grad f \cdot d\rr \end{equation} we obtain the standard product rule \begin{equation} \grad(fg) = g\,\grad{f} + f\,\grad{g} \end{equation}

Similar rules can be derived for other operators. For instance, if $f\in\bigwedge^0$ and $\alpha\in\bigwedge^1$, then \begin{equation} {*}d(f\,\alpha) = {*}(df\wedge\alpha+f\,d\alpha) = {*}(df\wedge\alpha) + f\,{*}d\alpha \end{equation} and recalling that \begin{align} {*}(F\wedge G) &= (\FF\times \GG) \cdot d\rr \\ {*}dF &= (\grad\times\FF) \cdot d\rr \end{align} we see that \begin{equation} \grad\times(f\GG) = \grad f\times\GG + f\,\grad\times\GG \end{equation} Similarly, \begin{equation} {*}d{*}(f\,\alpha) = {*}d(f\,{*}\alpha) = {*}(df\wedge{*}\alpha+f\,d{*}\alpha) = {*}(df\wedge{*}\alpha) + f\,{*}d{*}\alpha \end{equation} and recalling that \begin{align} {*}(F\wedge {*}G) &= \FF\cdot \GG \\ {*}d{*}F &= \grad\cdot\FF \end{align} we see that \begin{equation} \grad\cdot(f\GG) = \grad f\cdot\GG + f\,\grad\cdot\GG \end{equation}

The product rules considered so far are standard results in vector calculus. Here's one which may be less familiar. Suppose $\alpha,\beta\in\bigwedge^1$. Then \begin{align} {*}d{*}\left({*}(\alpha\wedge\beta)\right) &= {*}d(\alpha\wedge\beta) \nonumber\\ &= {*}(d\alpha\wedge\beta - \alpha\wedge d\beta) \nonumber\\ &= {*}(\beta\wedge d\alpha - \alpha\wedge d\beta) \nonumber\\ &= {*}(\beta\wedge {*}{*}d\alpha - \alpha\wedge {*}{*}d\beta) \nonumber\\ &= {*}(\beta\wedge {*}({*}d\alpha)) - {*}(\alpha\wedge {*}({*}d\beta)) \end{align} We have shown that \begin{equation} \grad\cdot(\FF\times\GG) = \GG\cdot(\grad\times\FF) - \FF\cdot(\grad\times\GG) \end{equation}