I would appreciate if someone help me to solve the following problem. If I have copula $C(u_1,\ldots,u_n)$ it is obvious how to obtain copula density. It is $$c(u_1,\ldots,u_n)=\frac{\partial^n C(u_1,\ldots,u_n)}{\partial u_n\ldots \partial u_1}\,.$$
Does the previous equation hold in case when copula function is replaced with survival copula $\hat{C}?$ Is this correct
$$c(u_1,\ldots,u_n)=(-1)^n\frac{\partial^n \hat{C}(u_1,\ldots,u_n)}{\partial u_n\ldots \partial u_1}\,.$$ This is just my trial, I have no the proof.