Abstract Given a function f : ( 0 , ∞ ) → R and a positive semidefinite n × n matrix P, one may define a trace functional on positive definite n × n matrices as A ↦ Tr ( P f ( A ) ) . For differentiable functions f, the function A ↦ Tr ( P f ( A ) ) is differentiable at all positive definite matrices A. Under certain continuity conditions on f, this function may be extended to certain non-positive-definite matrices A, and the directional derivatives of Tr ( P f ( A ) ) may be computed there. This note presents conditions for these directional derivatives to exist and computes them. These conditions hold for the function f ( x ) = log ⁡ ( x ) and for the functions f p ( x ) = x p for all p > − 1 . The derivatives of the corresponding trace functionals are computed here, and an alternative derivation of the directional derivatives using integral representations is provided.