For the inductive step, we know by the product rule for two functions that

\begin{equation*} (f_1f_2f_3 \cdots f_k f_{k+1})' = (f_1f_2f_3\cdots f_k)'f_{k+1} + (f_1f_2f_3\cdots f_k)f_{k+1}'\text{.} \end{equation*}

Then use the inductive hypothesis on the first summand, and distribute.