In this paper, new three level implicit finite difference methods of O(k^2+h^2) and O(k^2+h^4) are proposed for the numerical solution of fourth order quasi-linear parabolic partial differential equations in one space variable, where k\gt 0 and h\gt 0 are grid sizes in time and space coordinates respectively. In both cases, we use only nine grid points. The numerical solution of \partial u/\partial x is obtained as a by-product of the method. The characteristic equation for a model problem is established. Application to a linear singular equation is also discussed in detail. Four examples illustrate the utility of the new difference methods.