It seems to me to follow from the definition of a partial derivative. If f(x,y) = fx + fy, then what else can Δf be other than ∂f/∂x*Δx + ∂f/∂y*Δy? Then in the limiting case, the changes in f, x and y becomes differentials instead. All this seems to be given by the definitions themselves, is it not? So we have df = ∂f/∂x*dx + ∂f/∂y*dy All that's left is dividing by the parameter differential dt and we have our chain rule. I seem to recall there being entire lectures devoted to 'proving' this however. What am I missing?