Question on Human Capital Accumulation Model by Acemoglu/Autor I was reading the lecture notes by Acemoglu and Autor on Labour Economics. https://economics.mit.edu/files/4689

I was able to understand equations 1.1 to 1.8 of the notes; however, I am unable to understand how equations 1.9 and 1.10 were derived from equations 1.8 and 1.9 respectively. I understand that integration and differentiation was applied, but I am not able to figure out the exact steps involved.

For the transition from equation 1.8 to 1.9, I tried to understand in terms of general power rule and general exponential rule of integration, but there are so many terms involved that I am unable to think how these rules apply.

For the transition from 1.9 to 1.10, I tried to understand in terms of general product rule and quotient rule of differentiation, but I am unable to figure out exact steps.

I would deeply appreciate if anyone could help me understand these steps. Please let me know if any other info is required from me.

Thank you.
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 $