Find all continuous functions $ f : \mathbb{R} \to \mathbb{R} $ such that $(f(x)g(x))' = f'(x)g'(x) $ 
Find all continuous functions $ f : \mathbb{R} \to \mathbb{R} $ such that $(f(x)g(x))' = f'(x)g'(x), f,g \neq const $    

My solution: $ f'(x)g(x)+g'(x)f(x) = f'(x)g'(x)     \\
g(x)+g'(x)f(x)/f'(x) = g'(x)      \\
f(x)/f'(x) = (g'(x)-g(x))/g'(x) $ 
But how to proceed further to reduce to something concrete, I do not know

You can fix a function $f(x)$, and then you obtain a differential equation for $g(x)$, which has a unique solution up to a constant multiple:
$$
f'(x)g(x) + g'(x)f(x) = f'(x)g'(x)
$$ 
$$
g'(x) = \frac{g(x)}{1 - \frac{f(x)}{f'(x)}}
$$
