  We use a low redshift expansion of the cosmological equations of extended
(scalar-tensor) quintessence to divide the observable Hubble history parameter
space in four sectors: A forbidden sector I where the scalar field of the
theory becomes imaginary (the kinetic term becomes negative), a forbidden
sector II where the scalar field rolls up (instead of down) its potential, an
allowed `freezing' quintessence sector III where the scalar field is currently
decelerating down its potential towards freezing and an allowed `thawing'
sector IV where the scalar field is currently accelerating down its potential.
The dividing lines between the sectors depend sensitively on the time
derivatives of the Newton's constant G over powers of the Hubble parameter. For
minimally coupled quintessence which appears as a special case for a constant G
our results are consistent with previous studies. Observable parameter \chi^2
contours based on current data (SNLS dataset) are also constructed on top of
the sectors, for a prior of \Omega_m=0.24. By demanding that the observed
2\sigma \chi^2 parameter contours do not lie entirely in the forbidden sectors
we derive stringent constraints on the current second time derivative of
Newton's constant G. In particular we find {\ddot G}/G >-1.91 H_0^2=-2
10^{-20}h^2 yrs^{-2} at the 2\sigma level which is complementary to solar
system tests which constrain only the first derivative of G as |{\dot
G}/G|<10^{-14}yrs^{-1} at 1\sigma.
