Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on the Lagrangian of a free particle. At the very top of page 7, he equates two Lagrangian by
$$\mathrm{L}'=\mathrm{L}(v'^2)=\mathrm{L}(v^2+2v \cdot \epsilon+ \epsilon^2).$$
Then he goes to to say that you can expand this in powers of $ \epsilon$ and neglect all terms above the first order to get
$$\mathrm{L}(v'^2)=\mathrm{L}(v^2)+\frac{\partial \mathrm{L}}{\partial v^2}2v \cdot \epsilon.$$
How do you expand in terms of $\epsilon$ to get this result? This Lagrangian seems to me to be a function of three variables and I am looking at the multivariable Taylor expansion and I am not seeing how Landau could get his result.