I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below: $$ \bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm2}[\boldsymbol\phi]_\times+\frac{\Vert\boldsymbol{\phi}\Vert-\sin\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm3}[\boldsymbol\phi]_\times^\rm2 $$ where $\boldsymbol\phi\in\mathbb{R}^3$ is parameter space as called axis-angle representation, $[\boldsymbol\phi]_\times \in \frak{so}(3)$ is lie algebra of $SO(3)$.
I found a lot of materials but they don't gives specific derivation or development of lemmas.
I have two questions with proof of above equation.
Question #1
p.20 of "The Optimal Kinematic Design of Mechanisms" by FC Park provides Lemma 2.1. But his proof doesn't provide detailed transfer between equalities. For example,
$$ \begin{aligned} \frac{d}{dt}e^{x(t)}&=\lim_{h\to0}\frac{1}{h}(e^{x(t+h)}-e^t)\\&=\lim_{h\to0}\frac{1}{h}(e^{x(t)+h\dot{x}(t)+O(h^2)}-e^{x(t)}) \Leftarrow\text{Taylor expansion} \\&=\lim_{h\to0}\frac {1}{h}(e^{x(t)}+{\it h\frac{d}{dh}}e^{x(t)+h\dot{x}(t)+O(h^2)}-e^{x(t)})&\Leftarrow ???\\&=\lim_{h\to0}\frac{d}{dh}e^{x(t)+h\dot{x}(t)} \end{aligned} $$ I didn't understand how the equal sign was included in the third line of the development of equation.
Question #2.
I don't understand above development. Nevertheless, I wanted to at least partially understand the proof, so I searched YouTube, found out that Sneddon's formula was used in Park's proof, and looked for a related proof on StackExchange.
In StackExchange, @user8268 derive specific(set $t=1$) form of Sneddon's formula as below:
$$ \begin{aligned} J(1)&=\frac{d}{dϵ}\exp(V+ϵW)|_{ϵ=0}=\lim_{n→∞}\frac{d}{dϵ}(1+\frac{V+ϵW}{n})^n|_{ϵ=0}\\&=\lim_{n→∞}∑_{k=0}^{n}(1+\frac{V}{n })^kW(1+\frac{V}{n})^{n−k−1}\\&=∫^1_0\exp(sV)W\exp((1−s)V)ds \\&=(∫^1_0\exp(sV)W\exp(−sV)ds)\exp(V). \end{aligned} $$
I think it's correct to use the quadrature method when moving from the second to the third line, but I'm not sure how to set the variable. The development of this formula is the question.