If $j$ is a continuous variable, then differentiating the function $f(j)=j(j+1)$ with respect to $j$ gives $f'(j)=2j+1$. Of course I've chosen the letter to evoke quantum mechanical angular momentum, in which case for integer or half-integer values of $j$ we can interpret these two expressions as the eigenvalue of the squared angular momentum and the multiplicity of the angular momentum.
Is there any nice interpretation of this? As I was looking at the automatically generated list of "Questions that may already have your answer," I came across a comment that asks exactly the same question.
One reason to believe that it has no very special interpretation is that since the actual variable is discrete, the derivative $f'$ would really represent an approximation to a divided difference, and the relevant difference for a unit change in $j$ does not necessarily equal the derivative unless you evaluate the derivative at the correct place.
It seems to me that there is a second reason not to expect anything special here, which is that the correspondence doesn't seem to work except in three dimensions. For a rotor in $d$ dimensions, the eigenvalue of the squared angular momentum operator is $j(j+d-2)$. I don't know what the multiplicity of states is in general, but I suppose it's a polynomial of order $d-2$. E.g., for $d=2$, the multiplicity is 2 ($m=\pm j$), which doesn't equal the derivative of $j(j+d-2)=j^2$. On the other hand, I guess it's possible that there is a nice interpretation, and the nice interpretation tells us that there's something special about three dimensions.
Related: What is known about the hydrogen atom in $d$ spatial dimensions?
