In here, they describe the covariant derivative of a quantity $A^\mu B^\mu$. As i understand, scalar products are just like inner products
But we already defined the covariant derivative of a vector field with near identical components $V^a e_a$. And to my understanding, the definition of $\nabla_b V^a$ is the component form of the derivative. So how does it make sense we are taking derivative again?
The covariate derivative of a scalar along a vector field is simply its derivative along that vector field. Morally speaking, the covariate derivative of an inner product of vector fields should obey some kind of product rule relating it to the covariate derivatives of the vector fields. And in fact that is what the fundamental theorem of Riemannian geometry says.