I am implementing higher order derivatives for FEM. Example, to solve a Poisson problem, biharmonic or triharmonic PDE one needs first, second or third order derivatives respectively. As usually done in FEM, there is a mapping from a reference element to a physical element e.g. $F : \hat{K} \rightarrow K$ with the associated coordinates $(\hat{x}, \hat{y})$ and $(x,y).$ If we assume a linear affine map on a quadrilateral mesh, then for a function $u,$ the derivative is easily computed and given by $\nabla u = J^{-T}  \nabla \hat{u}.$  Now, my aim is to compute $\nabla^2 u, \nabla^3 u,...$ possibly $\nabla^n u.$I will appreciate any ideas on how to compute these derivatives. 
implementing higher order derivatives for finite element
finite element
You appear to have pretty much answered your own question.Changing notation slightly to use $\bf x$ for physical coordinates and $\bf \xi$ for the reference coordinate (because I have a nasty habit of leaving tildes off variables where I shouldn't) you have in component form that $\frac{\partial}{\partial x_{i}}u\left(\bf{x}\right)=\sum_{j}\frac{\partial\xi_{j}}{\partial x_{i}}\frac{\partial}{\partial\xi_{j}}u\left(\bf{\xi}\right)$so using the definition $\frac{\partial^2 f}{\partial x_i^2} := \frac{\partial}{\partial x_i}\frac{\partial f}{\partial x_i}$we get$\frac{\partial^2 u}{\partial x_i^2} = \frac{\partial}{\partial x_i}\frac{\partial u}{\partial x_i} = \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_{i}}u\left(\bf{x}\right)= \frac{\partial}{\partial x_i} \sum_{j}\frac{\partial\xi_{j}}{\partial x_{i}}\frac{\partial}{\partial\xi_{j}}u\left(\bf{\xi}\right) = \sum_k \frac{\partial\xi_{k}}{\partial x_{i}}\frac{\partial}{\partial\xi_{k}}\sum_{j}\frac{\partial\xi_{j}}{\partial x_{i}}\frac{\partial}{\partial\xi_{j}}u\left(\bf{\xi}\right)$.Assuming your mapping is linear, we can take the derivative through the summation and write$\frac{\partial^2 u}{\partial x_i^2} = \sum_{j,k} \frac{\partial\xi_{k}}{\partial x_{i}}\frac{\partial\xi_{j}}{\partial x_{i}}\frac{\partial^2 }{\partial \xi_k \partial\xi_{j}}u\left(\bf{\xi}\right)$.Hopefully you see the connection to what you're already doing with first derivatives.