Chapter 5, Problem 32SE

(Requires calculus) Use mathematical induction and the productrule to show that ifnis a positive integer and $f_1\left(x\right),f_2\left(x\right),\mathrm{...}{f}_n\left(x\right)$are all differentiable functions, then

   $\begin{array}{c}\frac{\left(f_1\left(x\right)f_2\left(x\right)\mathrm{...}f\left(x\right)\right)\text{'}}{f_1\left(x\right)f_2\left(x\right)\mathrm{...}{f}_n\left(x\right)}\\ =\frac{f_1^{\text{'}}\left(x\right)}{f_1\left(x\right)}+\frac{f_2^{\text{'}}\left(x\right)}{f_2\left(x\right)}+\mathrm{...}+\frac{f_n^{\text{'}}\left(x\right)}{f_n\left(x\right)}\end{array}$