On the Higher-Order Derivatives of Spectral Functions: Two Special Cases
  In this work we use the tensorial language developed in [8] and [9] to differentiate functions of eigenvalues of symmetric matrices. We describe the formulae for the k-th derivative of such functions in two cases. The first case concerns the derivatives of the composition of an arbitrary differentiable function with the eigenvalues at a matrix with distinct eigenvalues. The second development describes the derivatives of the composition of a separable symmetric function with the eigenvalues at an arbitrary symmetric matrix. In the concluding section we re-derive the formula for the Hessian of a general spectral function at an arbitrary point. Our approach leads to a shorter, streamlined derivation than the original in [6]. The language we use, based on the generalized Hadamard product, allows us to view the differentiation of spectral functions as a routine calculus-type procedure. 
Hristo S. Sendov