  Equations related to the Polchinski version of the exact renormalisation
group equations for scalar fields which extend the local potential
approximation to first order in a derivative expansion, and which maintain
reparameterisation invariance, are postulated. Reparameterisation invariance
ensures that the equations determine the anomalous dimension eta unambiguously
and the equations are such that the result is exact to O(epsilon^2) in an
epsilon-expansion for any multi-critical fixed point. It is also
straightforward to determine eta numerically. When the dimension d=3 numerical
results for a wide range of critical exponents are obtained in theories with
O(N) symmetry, for various N and for a ranges of eta, are obtained within the
local potential approximation. The associated eta, which follow from the
derivative approximation described here, are found for various N. The large N
limit of the equations is also analysed. A corresponding discussion is also
given in a perturbative RG framework and scaling dimensions for derivative
operators are calculated to first order in epsilon.
