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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter

I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I quote from his answer without modifying anything)

"In general, the distance function has one-sided directional derivatives everywhere. This derivative has a nice description in the case when you fix $p\in M$ and study the function $f=d(p,\cdot)$. Namely let $q\in M$, $q\ne p$, and denote by $\vec{qp}$ the set of initial velocity vectors (in $T_qM$) of unit-speed minimizing geodesics from $q$ to $p$. Then, for a vector $v\in T_qM$, the one-sided derivative $f'_v$ of $f$ in the direction of $v$ is $$ f'_v=\min\{-\langle v,\xi\rangle:\xi\in \vec{qp}\} . $$ This follows from the first variation formula and holds not only in Riemannian manifolds but also in Alexandrov spaces."

I'd like to find a book and read the proof of this part:

$$ f'_v=\min\{-\langle v,\xi\rangle:\xi\in \vec{qp}\} . $$

In his answer, he also stated: "but any book that covers Berger's lemma about geodesics realizing the diameter probably has directional derivatives as a sublemma", but an internet search on "Berger's lemma about geodesics realizing the diameter" wasn't helpful.

Could you please give me a reference for this result and perhaps also for Berger's lemma? Or if you could prove it, that'd be very helpful too! Thank you in advance!!