Derivatives of rotation number of one parameter families of circle   diffeomorphisms
  We consider the rotation number $\rho(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $\rho(t)$ is irrational $$\limsup_{t'\to t}(\rho(t')-\rho(t))/(t'-t)\geq 1.$$ 
Shigenori Matsumoto
math.DS