Lorentz invariance of an equation (metric)

[felixphysics]

 

[fzero]

I'm not sure what you mean by "metric is invariant." The metric transforms like a tensor, covariant or contravariant depending on whether the indices are up or down. The line element $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$ is Lorentz invariant, but that's not what appears in your formula.

You should write down the explict transformation rules for $V^\mu$, $\partial/\partial x^\mu$, $g^{\nu\sigma}$, and $g_{\nu\sigma}$. You will need to combine them in your expression and carefully apply the derivatives with the appropriate product rules for differentiation.