Lagrangian -> Equation of motion derivation

[Simonelis]

Homework Statement

I teach myself classical mechanics from David Tong
http://www.damtp.cam.ac.uk/user/tong/dynamics.html
From the homework set
I should verify that the Lagrangian

L=$\frac{1}{12}m^{2}\dot{x}^{4}+m\dot{x}^{2}V-V^{2}$

Yields the same equations as the mere L=$\frac{1}{2}m\dot{x}^{2}-V$

Homework Equations

Lagranges equation of motion

The Attempt at a Solution

This seems kinda trivial exercise, straightforward derivative computation yields

something like
$m\dot{x}^{2}\left(\frac{\partial V}{\partial x}-m\ddot{x}\right)-2V\left(\frac{\partial V}{\partial x}+m\ddot{x}\right)=0$

which should somehow factor out and give simple equation.

Still, it does not seem to...

Could anyone help with this one?

 

[vela]

I think you just made a sign error somewhere. Recheck your algebra.