Let $M$ be a riemannian manifold, and $\exp_p:B_0(\epsilon)\subset T_pM \to M$ be the exponential map. How to evaluate $d(\exp_p)_vv?$
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I assume that you, in $T_pM$, identify $v$ in the origin with $v$ in $T_vT_p M$. Then it's just the tangent to the geodesic starting in $p$ with initial vector $v$ at time $t=1$. This should follow more or less directly from the definition of $\exp_p$ -- you only need a curve passing through that point for which you can calculate the tangent, and said geodesic is the best canditate.
(The question is more difficult for $d(\exp_p)_v w$ if $w\neq v$, to caluclate that you need to study Jacobi fields with initial value $w$, if I recall correctly.)
Edit (adding a reference): This fact is also known as the Gauss Lemma, a reference would be Lemma 1.9.1 in Klingenbergs 'Riemannian Geometry', 2nd edition, de Gruyter. The remark about Jacobi fields is discussed in chapter 1.12 of that book.
As a further note, this fact is needed to prove the standard so called comparison theorems in Riemannian geometry, so any textbook adressing these will discuss this topic.