In Landau-Lifschitz, following expansion is given, We have, $$L(v'^2)=L(v^2+2\textbf{v}\epsilon+\epsilon ^2)$$$$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$ expanding this in powers of $\epsilon$ and neglecting powers of higher order, =$$L(v^2)+\frac{\partial L}{\partial v^2}2\textbf{v}.\epsilon$$$$L(v'^2)~\approx~L(v^2)+\frac{\partial L}{\partial (v^2)}2\textbf{v}\cdot\epsilon$$
L$L$ is Lagrangian. and $\textbf{v}'=\textbf{v}+\epsilon$. I am unable to follow what expansion this exactly is. It does not look like Taylor expansion.
