I am confused by this expansion in Landau and Lifshitz:
First, they define $\textbf{v}' = \textbf{v} + \textbf{$\epsilon$}$.
So for a function $L$,
$$L(v'^2) = L(v^2 + 2\textbf{v}\cdot\textbf{$\epsilon$} + \epsilon^2)$$
They then write an expansion, neglecting terms of order greater than one in $\textbf{$\epsilon$}$:
$$L(v'^2) = L(v^2) + \frac{\partial L}{\partial v^2}2\textbf{v}\cdot\textbf{$\epsilon$}$$
What confuses me is that it appears they are expanding about $v^2$, but they take a derivative with respect to $v^2$, not $v'^2$. How does this work?
This is atrocious notation. $L$ is a function of a scalar argument $x$. So they just mean the derivative
$$\left.\frac{\partial L}{\partial x}\right|_{x=v^2}.$$
They are then looking at the expansion evaluated at $\mathbf{v}\cdot\mathbf{\epsilon}+\mathbf{\epsilon}^2$ and preserving terms to first order in $\mathbf{\epsilon}$.