The covariant derivative of a metric is zero $g_{\alpha\beta;\sigma}=0$. Is the covariant derivative of a metric determinant zero following the assumption($g_{\alpha\beta;\sigma}=0$):
$$ g_{;\sigma}=g g^{\alpha\beta}g_{\alpha\beta;\sigma}=0? $$
In general, can the chain rule for ordinary derivative be used for the covariant derivative as
$$ F(f(x))_{;p}=F_{;p}f_{;p}? $$
Or always replace the covariant derivative of a scalar with the ordinary derivative?
