Why Must the Term dL/d(v^2) v.e Be Linear in v to Be a Total Time Derivative?

[rc75]

In section 4 of Landau and Lifgarbagez they derive the expression for the kinetic energy by expanding the Lagrangian around v+e. The resulting expression has a term which must be a total time derivative so that the equations of motion are unaffected. The text claims that the term dL/d(v^2) v.e must be linear in v to be a total time derivative, but I don't understand why this is.

 

[dslowik]

I just read that section.
I think it would have helped if they stated that the 2nd term is a total time derivative 'of a function of coordinates and time' ...
df(x,t) / dt = df/dx * dx/dt + df/dt (partial d's now)
Since f does not depend on the velocities, df/dx and df/dt don't, and the overall dependence of df/dt on v=dx/dt is linear.