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differentiation of Strain energy Function

I am having problem to find "Cauchy stress(σ) " and " Spatial elastic Tensor (c)" from the "strain energy function (ψ)" using 2nd Piola Kirchhoff stress ( S) of transversely isotropic hyper-elastic material. as per the Mechanics the 

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Derivatives of the invariants of a tensor

When you first start learning finite deformation plasticity, you will run into a plastic flow rate $ \ensuremath{\boldsymbol{d}}_p$ that can be derived from a flow potential $ \phi$ such that 

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