Differentiating both sides of an inequality with monotonic functions If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$? 
(Note: I've seen several questions asking the same thing without the condition of monotonicity, but the counterexamples given always involve a non-monotonic function, and it seems to me that this condition might be sufficient; I haven't been able to come up with any counterexamples myself.)
If not, is the stronger condition that $f^{(n)}(x)$ and $g^{(n)}(x)$ are monotone for either all natural $n$ or all $n\le N$ for some $N$ sufficient?

The question can be reformulated with $f(x)\le0$ and $f'(x)\le0$, without loss of generality.
Obviously, the growth of a function doesn't tell you its sign, as any added constant disappears by derivation.
Anyway, a starting function value can settle the case:
$$f'(x)\le0\land f(a)\le 0\implies\forall x\ge a: f(x)\le 0.$$
