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.



$\frac{d}{dt} (A+t B)^p\,\text{ for } p\geq 1$
In general, one standard approach would be to consider the map $A \mapsto A^p$ for any $p$. Then, actually write the map as $A \mapsto \exp(p \log A)$. Then, the derivative that you are after is the Fréchet derivative in direction $B$. This does not look entirely clean, but at this point it is just the chain rule (e.g., see [the derivative of the matrix exponential here](https://en.wikipedia.org/wiki/Matrix_exponential), and of the [matrix logarithm (see Slide 38)](http://www.6ecm.pl/docs/Higham.pdf); also in the same slides, you'll see numerical procedures for computing these derivatives numerically efficiently).


Alternatively, check out the [Daleckii-Krein theorem](https://books.google.com/books?id=KHrHCQAAQBAJ&lpg=PA120&ots=SC2Eo-P2WZ&dq=daleckii%20krein%20theorem&pg=PA120#v=onepage&q=daleckii%20krein%20theorem&f=false), which will show you how to obtain "nice" closed forms for derivatives of matrix functions similar to those in the above question.


