  We study the effect of left-right symmetry on the distribution of the
parametric derivative of the dimensionless conductance T with respect to an
external parameter X, dT/dX, of ballistic chaotic cavities with two leads, each
supporting N propagating modes. We show that T and dT/dX are uncorrelated for
any N. For N=1 we calculate the distribution of dT/dX in the presence and
absence of time-reversal invariance. In both cases, it has a logarithmic
singularity at zero derivative and algebraic tails with an exponent different
from the one of the asymmetric case. We also obtain explicit analytical results
for the mean and variance of the distribution of dT/dX for arbitrary N.
Numerical simulations are performed for N=5 and 10 to show that the
distribution P(dT/dX) tends towards a Gaussian one when N increases.
