The way to think about Leibniz notation is that there's a manifold M, and all the quantities f,g,x,y… are functions from M to the reals. Then you can talk about

The way to think about Leibniz notation is that there's a manifold M, and all the quantities f,g,x,y… are functions from M to the reals. Then you can talk about the derivative of any of these quantities with respect to any of the others, without thinking of any of them as the arguments of any of the others.