Reformulation of the Gaussian error propagation for a mixture of dependent and independent variables

The Gaussian error propagation is a state of the art expression in error analysis for estimating standard deviation for an expression f(x1,...,xn,z) via its variables. One of its basic assumptions is the independence of the measurable variables in its argument. However, in practice, measurable quantities are correlated somehow, and sometimes, z depends on some of the xi's. We provide the generalized version of the Gaussian error propagation formula in this case. We will prove this with the formula for total derivative of a general multivariable function for which some of its variables are not independent from the others; a counterpart to the probability approach of this subject.