Am trying to solve some differentials from a paper called "Euler arc splines for curve completion" (https://pdfs.semanticscholar.org/e388/e6e12ba0292f31795a9a3eeca7595cd05779.pdf) as I'm trying to program up the algorithm that they describe. I've tried to keep the problem self-contained, but will edit the post if I've not provided enough information
My issue is when trying to calculate the partial differentials: $\frac{\partial \delta_k}{\partial s}$ and $\frac{\partial \delta_k}{\partial \eta}$ where $\eta = \kappa_1 s$ has been used as a substitution for the purpose of applying the algorithm.
The equation to differentiate is:
$\delta_k = - \frac{\lambda_1 V_k^x + \lambda_2 V_k^y}{2}$
where:
$\lambda_1 = \frac{2 (P_A^x + P_B^x + s \sum_{j=1}^{n}V_j^x) \sum_{j=1}^{n}(V_j^y)^2 - 2 (P_A^y + P_B^y + s \sum_{j=1}^{n}V_j^y) \sum_{j=1}^{n}(V_j^y V_j^x) }{ \sum_{j=1}^{n}(V_j^x)^2 \sum_{j=1}^{n}(V_j^y)^2 -(\sum_{j=1}^{n}(V_j^y V_j^x))^2}$
$\lambda_2 = \frac{2 (P_A^y + P_B^y + s \sum_{j=1}^{n}V_j^y) \sum_{j=1}^{n}(V_j^x)^2 - 2 (P_A^x + P_B^x + s \sum_{j=1}^{n}V_j^x) \sum_{j=1}^{n}(V_j^x V_j^y) }{ \sum_{j=1}^{n}(V_j^x)^2 \sum_{j=1}^{n}(V_j^y)^2 -(\sum_{j=1}^{n}(V_j^x V_j^y))^2}$
$V_j^x = \frac{2 sin(\Delta \theta_j) }{\Delta \theta_j} cos(\theta_A + \psi_j)$
$V_j^y = \frac{2 sin(\Delta \theta_j) }{\Delta \theta_j} sin(\theta_A + \psi_j)$
$\psi_j = \frac{\theta_{j-1} + \theta_j}{2} - \theta_0$
and
$\theta_i = \theta_0 + i \kappa_1 s + \frac{(i - 1) i}{2} s^2 \alpha$
Where I get particularly stuck is how to get the differentials with respect to $\eta$ given that it is a function of s. Am I overcomplicating it? I'm just not sure how to begin even typing this kind of problem into something like SageMath to work it out.
I'm guessing by back-substituting all of the equations, I can get $\delta_k$ in terms of s and $\kappa_1 s$ but then when differentiating, both seem to involve s, so I'm not sure what to do to terms that will include s but not $ \kappa_1 $.
What are the two partial differentials, and how do I get there?