Suppose I want to know how $f(x)g(x)$ changes when $x$ increases.
I compute $\frac{d(f(x)g(x))}{dx}=f'(x)g(x)+f(x)g'(x)$ using the product rule.
However the sign of $f'(x)g(x)+f(x)g'(x)$ turns out to be indeterminate: $f'(x)g(x)<0$ while $f(x)g'(x)>0$ and I know nothing about the magnitudes of these functions.
If I decide to then compute $f'(x)$ and $g'(x)$ independently, will I always get that one of the functions is increasing in $x$ while the other is decreasing in $x$?
No, because your values also depend on whether the values of f(x) and g(x) are above the x axis or not.