Let be a pair of Hermitian operators in n and A = A1 + iA2. We investigate certain differential properties of the numerical range map with the aim of better understanding the nature of the numerical range W(A) of A. For example, the joint eigenvalues of A correspond to the stationary points of (i.e. points where the derivative vanishes). Moreover, points x where rank get mapped by into the interior W(A)° of W(A). For n = 2, it turns out that if A1 and A2 have no common invariant subspace, then the image under of the set Σ1(A) consisting of those points x with rank is precisely the boundary ∂W(A) of W(A), and the image under of the rank 2 points for is precisely W(A)°; there are no rank 0 points for . As a consequence (for n = 2) we have that A1A2 = A2A1 iff .