= Solution
For $v$ near $0\in T_pM$, let $\gamma_v$ be the unique geodesic with $\gamma_v(0)=p$ and $\dot\gamma_v(0)=v$. The <exponential map> is $\exp_p(v)=\gamma_v(1)$.
The constant initial velocity $0$ gives $\exp_p(0)=p$. Varying the initial velocity through $sv$ yields $\exp_p(sv)=\gamma_v(s)$, so
$$
(d\exp_p)_0(v)=\left.\frac d{ds}\right|_{0}\exp_p(sv)=v.
$$
Thus $(d\exp_p)_0$ is the identity. The <inverse function theorem> makes $\exp_p$ a diffeomorphism from a neighborhood of $0$ onto a neighborhood of $p$. Coordinates from an orthonormal basis of $T_pM$ are the <geodesic normal coordinates>.
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