Table 2.6.5(a) contains the solution. Note that the outcome is consistent with the comment about the pattern for the Product rule stated in Section 2.3.$$

Take the natural log of both sides. The log of the product on the right is the sum of logs. Differentiate both sides. On the left, the derivative of $F$ is $F\prime \/F$ while on the right, the derivatives of the logs yield a sum of fractions of the form derivative divided by function. Multiply through by $F$, which is itself the original product, and simplify.