Differential of the Trace for functions applied to unbounded self-adjoint operators

Question

Let $f\colon\mathbb{R}\to\mathbb{R}$ be in the set of Schwartz functions (or any functions with nice enough integrating properties on the real axis), $A,M$ two (possibly unbounded) self-adjoint operators on a separable Hilbert space $\mathcal{H}$. Assume that:

1. $A+M$ is self-adjoint.
2. The operators $f(A+M),f(A),f(M),f'(A)M$, which are defined by functional calculus, are trace class.

Is it true that the Frechet derivative of $A\mapsto Tr(f(A))$ at $A$ applied to the operator $M$ is $Tr(f'(A)M)$ ?

My research: It resembles the formulae obtained using the spectral shift function, however the perturbation $M$ may not be bounded/trace class/Hilbert-Schmidt, even though $f(M)$ can, and the spectral shift function usually does not have an analytical expression. For nice enough functions $f$, I would assume this result to hold.