In a first part, we shall prove a quantitative nonvanishing result conjectured by Ph.Michel and A.Venkatesh which concerns the special derivatives occuring in the Gross-Zagier formula. Our method relies on two classical equidistribution Theorems in arithmetic geometry. In a second part, we shall explain how to refine these results with methods from analytic number theory. We shall emphasize on the new aspects of this analytic approach which may hopefully be used in other contexts.