Let $(M^n,g)$ be a Riemannian $n$-manifold. Let $p\in M$, and let $v\in T_pM$. By the existence and uniqueness theorem (of ODEs, hence of geodesics), there is a unique geodesic $\gamma$ on $M$ such that $\gamma(0)=p$ and $\gamma'(0)=v$. The exponential map at $p$, denoted by $exp_p:T_pM\to M$, is then defined by \begin{align} exp_p(v)=\gamma_v(1) \end{align} It is standard that the differential of $exp_p$ at $0\in T_pM$ is the identity map on $T_pM$, after identifying $T_0(T_pM)\simeq T_pM$.
In this question I'm interested in the following map $\mathcal{E}:TM\to M$ defined by \begin{align} \mathcal{E}(p,v):=exp_p(v) \end{align} for any $p\in M$ and any $v\in T_pM$. This is sort of a 'global' version of exponential map, and I'm interested in:
Compute the differential of $\mathcal{E}$ at any point $(p,v)\in TM$.
If $\{x^i\}_{i=1}^n$ are local coordinates of $p$ and $\{a^i\}_{i=1}^n$ are local coordinates of $v$: \begin{align} v=a^i\frac{\partial}{\partial x^i}\bigg|_p \end{align} then one of the greatest difficulty I have is that I have no idea how to compute the partial derivatives $\frac{\partial E}{\partial x^i}$ since I have no idea how $exp_p$ depends on $p$. Actually I'm also not quite sure about the other part: the partial derivatives $\frac{\partial E}{\partial a^i}$. I suspect Jacobi fields may come in handy but I'm not sure how.
Any comment, hint or answer is greatly welcomed and appreciated.