A rational map f on the Riemann sphere is said to be postcritically finite if all of its critical points have finite forward orbits. In the 1980s, William Thurston proved that such a map f induces a holomorphic endomorphism on an associated Teichmueller space, and he showed that this endomorphism has a unique fixed point. In this talk, we study the derivative of this endomorphism at the unique fixed point. We focus on the case where f is a quadratic polynomial with a periodic critical point and prove some surprising things about this derivative of the associated endomorphism on Teichmueller space.