Confusion between covariant and partial derivatives

Question

Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will remain constant - meaning thar it will be parallel transported along X axis, and its covariant derivative will be zero. But what about its partial derivative (or just derivative, since its a 1d case)?