# The derivative of a product of more than two functions

I'm trying to generalize the product rule to more than the product of two functions using the fact that I can treat the product of $n$-1 functions as a single one. Here is an example of what I mean:

$[f(x)g(x)h(x)]' = [f(x)p(x)]'$ where $p(x) = g(x)h(x)$

$[f(x)p(x)]' = f'(x)p(x) + f(x)p'(x) = f'(x)p(x) + f(x)[g(x)h(x)]'$

$f'(x)p(x) + f(x)[g(x)h(x)]' = f'(x)g(x)h(x) + f(x)[g'(x)h(x) + g(x)h'(x)]'$

which equals $f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)$

I generalized this as follows:

$$\Big[\prod_{i=1}^{n}f_i(x)\Big]'= f_1'(x)g_1(x) + f_1(x)g'_1(x)$$