 Ordinary covector derivative under nonlinear coordinate changes

Let $A_b^{\prime }=J^d{}_b{A}_d$, where $J^d{}_b=\partial x^d/\partial x^{\prime b}$. ThenThe second term prevents the ordinary derivative from being a generally covariant tensor. It vanishes for affine coordinate changes, including Cartesian Lorentz transformations, so the restricted claim of Lorentz covariance is valid. Antisymmetrization cancels the same term for arbitrary smooth coordinate changes and gives the exterior derivative of a one-form.