In Landau-Lifschitz, following expansion is given, 
We have,
$$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$
expanding this in powers of $\epsilon$ and neglecting powers of higher order,
$$L(v'^2)~\approx~L(v^2)+\frac{\partial L}{\partial (v^2)}2\textbf{v}\cdot\epsilon$$

$L$ 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.