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Just reading the first book by Landau in the theoretical physics course, and I need some guidance about one point (cf. 4, Lagrangian for a free particle.) Notations: L and L' are Lagrangians referred to different inertial frames of reference. e is a element of velocity between L and L'.

It says " We have L' = L (v'^2) = L (v^2 + 2ve + e^2 ). Expanding this expression in powers of e and neglecting the terms above the first order, we obtain

L (v'^2) = L (v^2) + partial derivative of L respect to v^2 times 2ve

So, if this is supossed to be a Taylor serie with n from zero to one, why appears the derivate of L with respect to v^2. Can someone, please, make all steps explicit?

Any help would be very appreciated!