Answer
$$\operatorname{div}(\nabla f \times \nabla g)=0$$
Work Step by Step
Since \begin{align*}
\nabla f&=\frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}+ \frac{\partial f}{\partial z} \\
\nabla g&= \frac{\partial g}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial g}{\partial z}
\end{align*}
Then
\begin{align*}
\operatorname{div}(\nabla f \times \nabla g)&=\begin{array}{|||}
\frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\
\frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}& \frac{\partial f}{\partial z}\\
\frac{\partial g}{\partial x}& \frac{\partial g}{\partial y}& \frac{\partial g}{\partial z}
\end{array} \\
&= \frac{\partial}{\partial x}\left( \frac{\partial f}{\partial y} \frac{\partial g}{\partial z}- \frac{\partial f}{\partial z} \frac{\partial g}{\partial y} \right) - \frac{\partial}{\partial y}\left( \frac{\partial f}{\partial x} \frac{\partial g}{\partial z}- \frac{\partial f}{\partial z} \frac{\partial g}{\partial x}\right) + \frac{\partial}{\partial z}\left( \frac{\partial f}{\partial x} \frac{\partial g}{\partial y}- \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} \right) \\
&= \left( \frac{\partial f}{\partial y} \frac{\partial^2 g}{\partial z\partial x}+ \frac{\partial^2 f}{\partial y\partial x} \frac{\partial g}{\partial z}- \frac{\partial f}{\partial z} \frac{\partial^2 g}{\partial y\partial x} - \frac{\partial f}{\partial z} \frac{\partial ^2g}{\partial y\partial x} \right) - \left( \frac{\partial f}{\partial x} \frac{\partial^2 g}{\partial z\partial y}+\frac{\partial^2f}{\partial x\partial y} \frac{\partial g}{\partial z}- \frac{\partial ^2f}{\partial z\partial y} \frac{\partial g}{\partial x}- \frac{\partial f}{\partial z} \frac{\partial^2 g}{\partial x\partial y}\right) + \left( \frac{\partial ^2f}{\partial x\partial z} \frac{\partial g}{\partial y}+\frac{\partial f}{\partial x} \frac{\partial^2g}{\partial y\partial z}- \frac{\partial^2 f}{\partial y\partial z} \frac{\partial g}{\partial x}- \frac{\partial f}{\partial y} \frac{\partial ^2g}{\partial x\partial z} \right) \\
&=0
\end{align*}
