If a differentiable function z ( x , y ) is one-to-one (injective) for each independent variable, e.g., z ( x , y ) is one-to-one for x at a fixed y while it is not necessarily one-to-one for ( x , y ) , then the following total differentials exist because each independent variable is a differentiable function for the other variables, e.g., x ( y , z ) .