I've followed a number of derivations of the coordinate expression for the covariant derivative of a one-form: $\nabla_\delta \omega_\alpha$. To summarize, one strategy is
We know how to take covariant derivatives of scalar functions and vector fields. Since the "product" of a one-form and a vector field is scalar, and we know how the covariant derivative behaves under "products", we can derive the covariant derivative for one-forms too.
So we start by examining $\nabla_u(\omega \otimes v)$: $$\nabla_u(\omega \otimes v) = \nabla_u \omega \otimes v + \omega \nabla_u \otimes v$$ or in coordinates $$\nabla_\delta(\omega_\alpha b^\beta) = (\nabla_\delta \omega_\alpha) v^\beta + \omega_\alpha (\nabla_\delta v^\beta)$$ $$= (\nabla_\delta \omega_\alpha) v^\beta + \omega_\alpha (\partial_\delta v^\beta) + \omega_\alpha \Gamma^\beta_{\gamma\delta}v^\gamma$$
Contracting on $\alpha$ and $\beta$ on both sides we get $$\nabla_\delta(\omega_\alpha v^\alpha) = (\nabla_\delta \omega_\alpha) v^\alpha + \omega_\alpha (\partial_\delta v^\alpha) + \omega_\alpha \Gamma^\alpha_{\gamma\delta}v^\gamma.$$
The left hand side $\omega_\alpha v^\alpha$ is a scalar function, so $$\nabla_\delta(\omega_\alpha v^\alpha) = (\partial_\delta \omega_\alpha)v^\alpha + \omega_\alpha(\partial_\delta v^\alpha)$$ so with the appropriate cancelling I can get my answer. Why was I allowed to treat the product $\omega_\alpha v^\alpha$ as a scalar function when I differentiated? I understand that the action of a one-form on a vector field is a scalar function, but I expanded this as some kind of product. It's not a tensor product of $\omega$ and $v$, rather a contracted tensor product. Should I think of this as just a set of $n^2$ derivatives? What is the correct interpretation of the covariant derivative of $\omega_\alpha v^\alpha$?