On directional derivatives of trace functionals of the form $A\mapsto\Tr(Pf(A))$

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the function $A\mapsto \Tr(Pf(A))$ 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 \emph{directional} derivatives of $\Tr(Pf(A)$ 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.