Let x be a random variable with density function f and [a,b] be the possible set of values of x.
Let w(x) be a value associated with each value of x. I.e. w() is not known, and suppose that U(w(x)) is a function.
Find the derivative of H with respect to w over all x.
I don't know where to begin because I'm not even sure I understand the question.
I suspect the answer is f(x)U'(w(x)) only because it would be analogous to the case with finitely many possible value of x, but I dont know how to prove that.