Origin of the half factor in Euler-Lagrange for geodesics

[Alex Petrosyan]

I was wondering where does the 1/2 factor come from in the Euler-Lagrange equation, that is:
$$L = \sqrt{g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu}$$

implies that $$\partial_\mu L = \pm \frac{1}{2} (\partial_\mu g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu )$$

I'm not sure I entirely understand where it comes from. Intuitively, it might be because taking the partial derivative also is a contraction, but every book I've looked in, simply assumes this is true, and I don't understand how.

 

[Vanadium 50]

Doesn't it comes from the derivative of the square root?

 

[Alex Petrosyan]

@Vanadium 50 , that would also explain the sign ambiguity.