Differential of Exponential Map

Question

Let $(M, \langle-,-\rangle)$ be a Riemannian manifold, $p \in M$ and $v \in T_p M$. Let $\exp$ be the exponential map. How to prove $\langle (d \exp_p)_v (v), (d\exp_p)_v (v)\rangle = \langle v, v\rangle $? This is an important step in the proof of Gauss lemma and all the differential geometry books seem to think this is trivial, but I don't really see how to prove this.