  We analyse the implications of information about local derivatives from the
mass-radius diagram in neutron star matter. It is expected that the next
generation of gravitational wave and electromagnetic detectors will allow the
determination of the neutron star radius and mass with a small uncertainty.
Observations of neutron stars clustered around a given neutron star mass allow
the estimation of local derivatives in the $M(R)$ diagram, which can be used to
constrain neutron star properties. From a model-independent description of the
neutron star equation of state, it is shown that a $M(R)$ curve with a negative
slope at 1.4$M_\odot$ predicts a $2M_\odot$ neutron star radius below 12 km.
Furthermore, a maximum mass below 2.3$M_\odot$ is obtained if the $M(R)$ slope
is negative in the whole range of masses above $1M_\odot$, and a maximum mass
above 2.4$M_\odot$ requires the $M(R)$ slope to be positive in some range of
masses. Constraints on the mass-radius curve of neutron stars will place strong
constraints on microscopic models.
