Let $M$ be a Riemannian manifold and $p\in M$ a point. Let $E_{i}$ be an orthonormal basis of $T_{p}M$ and let $E\colon \mathbb{R}^{n} \to T_{p}M$ be the isomorphism sending $(x_{1},\ldots,x_{n})\mapsto \sum_{i=1}^{n}x_{i}E_{i}$. On a normal neighborhood $U$ of $p$ in $M$ (that is, the image under the exponential map $\exp_{p}$ of a star shaped open neighborhood of $0\in T_{p}M$) we can define a coordinate chart by $\varphi =E^{-1}\circ \exp_{p}^{-1}\colon U\to \mathbb{R}^{n}$. As with any other chart, the basis $\frac{\partial }{\partial x_{i}}$ of $T_{0}\mathbb{R}^{n}$ pulls back to a basis $\frac{\partial }{\partial x_{i} }|_{p}$ of $T_{p}M$.
Is this pullback the orthnormal basis that we started from?
My attempt: let $f\in C^{\infty }(M)$. We have to show that $E_{i}(f)=\frac{\partial }{\partial x_{i} }|_{p}(f)$. By definition:
$$\frac{\partial }{\partial x_{i} }|_{p}(f)=\frac{\partial( f\circ \exp_{p}\circ E)}{\partial x_{i} }(0)$$
I guess now I need to deal with the definition of the exponential map explicitly, but I am not sure on how to do this. So first $E$ sends $(x_{1},\ldots,x_{n})\mapsto \sum_{i=1}^{n}x_{i}E_{i}$. Then the exponential map sends a vector $v\in T_{p}M$ to $\gamma_{v}(1)$, where $\gamma_{v}$ is a geodesic with $\gamma_{v}(0)=p$ and $\dot{\gamma_{v} }(0)=v$. How do I use this now? I don't even know how geodesics look like without knowing the Riemannian metric $g$.
For example, the point $(1,0,\ldots,0)$ is sent first to $E_{1}$ and then we take the geodesic $\gamma$ with $\gamma(0)=p=(0,\ldots,0)$ (using the expression of the point in the normal coordinate chart) and $\dot{ \gamma}(0)=E_{1}$. Now $\gamma(1)$ should be the point in the trajectory of $\gamma $ at time 1, and I guess if we track $\gamma$ on $\mathbb{R}^{n}$ via the normal coordinates then this point should correspond to $(1,0,\ldots,0)$, but I am already not sure of these claims.
This also led me to the following question:
In a chart $(V,\psi)$ with $\psi(p)=0$, is it true that $\frac{\partial }{\partial x_{i} }|_{p}=\dot{ \gamma }(0)$ for any curve $\gamma $ in $M$ with $\gamma(0)=p$ and such that $\psi(\gamma)$ has speed $1$ in the positive direction of the $x_{i}$ axis?
EDIT (current thoughts on the questions):
I suspect now that the answer to the first question is positive and I suspect it will follow from the differential of the exponential map at $p$ being the identity on $T_{p}M$. But I still couldn't manage to write down a formal argument.
Regarding the second question, I still didn't find an answer and I still very interested in finding one. I think that there are at least two ways of defining the tangent space of $M$ at a point $p$, one with this abstract derivations and the other one by considering derivatives of curves in the manifold passing through $p$ at time $0$. I have never worked with this definition, but after thinking about the second question I got to the following conclusion: isn't that second question precisely the relation between these two definitions of the tangent space?