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?

Another more explicit example:
$$
f(x)=x-\frac\pi2+\arctan x,\quad g(x)=x.
$$
$$
f'(x)=1+\frac{1}{1+x^2},\quad g'(x)=1.
$$
