Derivative of a real-valued function w.r.t a vector?
It is a the vector comprising the derivatives of the function with respect to (w.r.t) each of the components of the vector.
As an example, consider the derivative of a particular scalar, d/da (a . b) = b. That is, the derivative of the "dot" product (a . b) w.r.t the vector a. If one thinks about the geometric definition of the dot product, then one can imagine the derivative giving a sense of how the scalar changes with respect to the vector (if still not clear, think now about d/da (a . a) = 2a).
Now, we can generalize this idea to the case a real-valued function, but remembering that this is a function of the elements of the vector about which we are differentiating.