  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.
