I've been thinking about something recently: The notation d2x/d2y actually represents something as long as x and y are both functions of some third variable, say u. Then you can take the second derivatives of both with respect to u and evaluate d2x/du2 × 1/(d2y/du2). Now I think it's also reasonable to express d2x/d2y as the product of dx/d and d/dy. Although dx/d is a notation I've never seen before I assume it's an operator. So how can the combination of two operators give an actual function? I think what I'm assuming is wrong so please explain why this does not work. If the notation dx/d isn't defined, why has no one defined it? If dx/d is not an operator, what is it? Because d/dy is certainly one, I've found it possible to calculate d/dx for various functions x and y by integrating the expression for d2x/d2y. There doesn't seem to be any sort of pattern, but I probably haven't looked hard enough.