There is a statement in both Wald and Carroll's GR texts that, in short, state that the derivative of a $(k,l)$-tensor is a $(k,l+1)$-tensor. In both places this as stated as though it should be obvious, but I am having trouble developing an intuition for this.
Could someone explain this? I am most comfortable thinking about tensors with the "slot machine" intuition, so if it could be explained in terms of "why the derivative increases the number of vector slots" that would be appreciated.
In short, these are directional derivatives, and the extra slot is for the vector that says what the direction is.