Chapter 3.9, Problem 86E

To determine

(a)

To show:

That $\frac{d}{dx}\ln \left(f\left(x\right)g\left(x\right)\right)=\frac{f^{\prime }\left(x\right)}{f\left(x\right)}+\frac{g^{\prime }\left(x\right)}{g\left(x\right)}$ such that $f$ and $g$ are differentiable functions.

To determine

(b)

To proof:

The new product rule if $\frac{{\left(f\left(x\right)g\left(x\right)\right)}^{\prime }}{f\left(x\right)g\left(x\right)}$ is equal to $\frac{d}{dx}\ln \left(f\left(x\right)g\left(x\right)\right)$.