  We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on
hyperbolic surfaces for which one disconnecting collar degenerates and prove
that $8\pi \nabla\log(\lambda_1)$ essentially agrees with the dual of the
differential of the degenerating Fenchel-Nielsen length coordinate. As a
consequence, we can improve previous results of Schoen, Wolpert, Yau and Burger
to obtain estimates with optimal error rates and obtain new information on the
leading order terms of the polyhomogeneous expansion of $\lambda_1$ of Albin,
Rochon and Sher.
