I am trying to understand the proof of the General Result for the Product Rule for Derivatives by reading this.
Relevant parts are as follows:
Basis for the induction $$ D_x \left({f_1 \left({x}\right) f_2 \left({x}\right)}\right) = D_x \left({f_1 \left({x}\right)}\right) f_2 \left({x}\right) + f_1 \left({x}\right) D_x \left({f_2 \left({x}\right)}\right) $$
Induction Hypothesis $$ D_x \left({\prod_{i=1}^k f_i \left({x}\right)}\right) = \sum_{i=1}^k \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \ne i} f_i \left({x}\right)}\right) $$
Induction Step $$ \begin{align} \tag{1} \kern-30pt D_x \left({\textstyle\prod\limits_{i=1}^{k+1} f_i \left({x}\right)}\right) &= D_x \left({\left({\textstyle\prod\limits_{i=1}^k f_i \left({x}\right)}\right) f_{k+1} \left({x}\right)}\right) \\ &= \tag{2} D_x \left({f_{k+1} \left({x}\right)}\right) \left({\textstyle\prod\limits_{i=1}^k f_i \left({x}\right)}\right) + D_x \left({\textstyle\prod\limits_{i=1}^k f_i \left({x}\right)}\right) f_{k+1} \left({x}\right) \\ &=\tag{3} D_x \left({f_{k+1} \left({x}\right)}\right) \left({\textstyle\prod\limits_{i=1}^k f_i \left({x}\right)}\right) + \left({\sum_{i=1}^k \left({D_x \left({f_i \left({x}\right)}\right) \textstyle\prod\limits_{j \ne i} f_i \left({x}\right)}\right)}\right) f_{k+1} \left({x}\right) \\ &= \tag{4} \sum_{i=1}^{k+1} \left({D_x \left({f_i \left({x}\right)}\right)\textstyle \prod\limits_{j \ne i} f_i \left({x}\right)}\right) \end{align} $$
Question
I am stuck at (3). How do I go from (3) to (4)? Specifically, what sort of algebraic manipulations need to be done and what are the motivations for doing those algebraic manipulations in order to arrive at (4)? To put it in another way, I would like to know that is the thought process that one goes through when simplifying (3) to (4).
Note: I hope someone can correct my LaTeX typesetting. I was under the impression that the align environment would automatically number the formulas I write. Thanks in advance.