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

$$f'(x)g(x)+f(x)g'(x)=f'(x)g'(x)$$
$$\frac{f(x)}{f'(x)}+\frac{g(x)}{g'(x)}=1$$
$$\frac{1}{[\ln f(x)]'}+\frac{1}{[\ln g(x)]'}=1$$
call $\ln f(x)=p(x)$ and $\ln g(x)=q(x)$
$$p'(x)=\frac{q'(x)}{q'(x)-1}=1+\frac{1}{q'(x)-1}$$
$$p(x)=x+\int\frac{1}{q'(t)-1}dt+c$$
