Consider the two dimensional sphere $\mathbb{S}^2$ and let $p, q \in \mathbb{S}^2$. Let $\text{exp}_{p}$ and $\text{exp}_{q}$ be the exponential maps on $\mathbb{S}^2$ at points $p$ and $q$ respectively. I am interested in the map $\psi := \text{exp}_{p}^{-1} \circ \text {exp}_{q}$ defined on the unit disc $\mathbb{D} \subset \mathbb{R}^2$. I expect that if $p$ and $q$ are nearby points, then the map $\psi$ is close to the identity map. My question is: Is there a way to quantify this closeness? More precisely, is it possible to write the Taylor series expansion of $\psi = (\psi_1, \psi_2)$ around the origin as \begin{align*} \psi_1(x,y) &= \psi_1(0) + \frac{\partial \psi_1}{\partial x}(0)~x + \frac{\partial \psi_1}{\partial y}(0)~ y + \ldots \\ \psi_2(x,y) &= \psi_2(0) + \frac{\partial \psi_2}{\partial x}(0)~x + \frac{\partial \psi_2}{\partial y}(0)~ y + \ldots \end{align*} and show that the only significant coefficients in the above expansion are $\frac{\partial \psi_1}{\partial x}(0) \approx 1$, $\frac{\partial \psi_2}{\partial y}(0) \approx 1$ and all other partial derivatives $\frac{\partial^{m+n} \psi_i}{\partial^{m} x \partial^{n} y}(0) \approx 0$, for all other choices of $m,n$ and any $i = 1,2$.
Any relevant reference will also be helpful.
Thanks!