I know the derivative of a CDF is the PDF, but what if you want to take a partial derivative. For example, I have a standard normal difference variable

$\displaystyle Y = (Xi-Xj)/(sqrt(Var(Xi) + Var(Xj)))$

I would like to take the derivative with respect to Xi. I am thinking this just invokes the chain rule and would give:

$\displaystyle PDF(Y) * (1/sqrt(VarI + VarJ))$

Is that correct?