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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(\...

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