The following is taken from (Applications of Lie Groups to Differential Equations by Olver): Consider two vector fields $v=\xi^i(x)\dfrac{\partial}{\partial x^i}$ and $w=\eta^i(x)\dfrac{\partial}{\partial x^i}$. Now his goal is to calculate the Lie derivative of $w$ with respect to $v$. He first calculates
$$ \left.w\right|_{\exp(\varepsilon v)x}=\left[\eta^i(x) + \varepsilon v(\eta^i)(x) + \mathcal{O}(\varepsilon^2) \right]\left.\dfrac{\partial}{\partial x^i}\right|_{\exp(\varepsilon v)x}. $$ So far so good, what I cannot calculate for the life of it is
$$ d \exp(-\varepsilon v)\left[ \left.w\right|_{\exp(\varepsilon v)x} \right], $$ any suggestions on how to calculate this quantity to order $\varepsilon$?