We know from calculus the product rule for differentiation $$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$$
A beginner who has not yet learned this may try doing something like this:
$$(f(x)g(x))' = f'(x)g'(x)$$
Now to my question, can we determine for which class of functions this will "accidentally" work? In other words what can we demand of $f,g$ for this to hold?