Given two positive self-adjoint operators $A,B$ on a Hilbert space. Let $p\geq 1$.

I would like to calculate $$\frac{d}{dt}|_{t=0} (A+tB)^p,$$

where the power is defined through the spectral theorem. The difficulty is that $A$ and $B$ do not commute in general. Is there a nice formula for that? I thought about using some integral representation of $x^p$, which hopefully would make the differentiation easier, but I was not successful.

**Update:**
Thanks for your answers. In my case, I actually only need it for $p\in [1,2)$. Using $y^p = \frac{\sqrt 3}{2\pi}\int_{0}^\infty dx \,\frac{y^2 x^{p-2}}{x+y}$, I thought about writing $C:=A+tB$ as $$C^p = \frac{\sqrt 3}{2\pi}\int_0^\infty C^2 x^{p-2} (x+C)^{-1} dx.$$
That might also work. I am still happy about other suggestions.