Calculating material jacobian matrix for ABAQUS UMAT subroutine

Although the methods for calculating jacobian matrix for well-known hyperelastic materials such as Neo-Hookean model is available online, there is no appropriate and comprehensive source for the more complicated formulations.

 One of the most famous formulations recently is following:

Strain Stored energy (W)=W(λi)

while λ is the eigenvalue of deformation gradient or principle stretch and n is the eigenvector of deformation gradient or principle direction 

then one can find the stress using following equation:

 σ=(λi ∂W/∂λi   ni ×ni) /J

while × is dyadic product and J is λ1 *λ2 *λ3 

Now I have the following questions: 

1- what is the explicit equation in terms λ to find jacobian matrix? 

2- do we need to subtract the hydrostatic pressure from the σ and then use new stress tensor and calculate material jacobian regarding new stress tensor?