I am currently writing a paper related to spotted owl conservation and reading a paper about demographic models for that species. It uses the Euler-Lotka equation, along with some facts about owl biology, to find a reduced characteristic equation $$\lambda ^ \alpha (\alpha-s/\lambda) = l_\alpha b,$$ where $\lambda$ is the population's growth rate, $\alpha$ Is the age of first reproduction, $s$ is the probability of an adult individual surviving to the next year, $l_a$ is the probability of a juvenile surviving to reproduction age, and $b$ is the number of offspring per capita. ADDITIONALLY, $l_\alpha = s'_0s_ds_1s$ (essentially the product of the probability it survives its dispersal times the probability it survives 2 more years—but this understanding is not necessary to the calculation).
I understand up to here, but then the authors embark on a sensitivity analysis for each of the variables using implicit partial differentiation, concepts which I think I understand fairly well. However, I cannot find the specific process the authors use to get this result: 
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Could someone please explain step-by-step how the authors reach these results. Thank you very much!
Edit: This is the work that I have so far for $\partial{\pi}$. $$\frac{\partial}{\partial{s_d}}[\lambda^\alpha(1-s/\lambda)] = \frac{\partial}{\partial{s_d}}[s'_0s_ds_1sb]$$ $$(\frac{\partial}{\partial{s_d}}[\lambda^\alpha])(1-s/\lambda) + \lambda^\alpha(-s\frac{\partial}{\partial{s_d}}[1/\lambda]) = s'_0s_1sb$$ $$\alpha\lambda^{\alpha-1}\frac{\partial{\lambda}}{\partial{s_d}}(1-s/\lambda) + \lambda^\alpha(-s*-1/\lambda^2\frac{\partial{\lambda}}{\partial{s_d}}) = s'_0s_1sb$$ Aside from factoring $\frac{\partial{\lambda}}{\partial{s_d}}$, I am stuck as to where to go from here.