Question on Human Capital Accumulation Model by Acemoglu/Autor
The FOC from (1.9) is (I'm skipping the content of the $\exp()$ expression for brevity):

$ \frac{1}{r+\nu+g_h + g_w} \left(\eta'(S)w(0) \exp(\cdot) + \eta(S) w(0) (r + \nu - g_w)\exp(\cdot)\right) \overset{!}{=} 0$

This makes use of product rule.
Since the first term can't be 0 due to the last equation on p.~11, it must be second one

$ \eta'(S)w(0) \exp(\cdot) = \eta(S) w(0) (r + \nu - g_w)\exp(\cdot) $

$ \eta'(S) = \eta(S) (r + \nu - g_w) $
yielding Eq. (1.10): 

$ \frac{\eta'(S)}{\eta(S)} = r + \nu - g_w $