Error Propagation and Error Partition Given a a general form of a model  $F$ ( which could be an explicit formula or  set of formulas) with its input $x$ and  output $y$.
Suppose that the input $x$ has an error $\delta x$, so that  given $ \tilde{x}=x+\delta x$ , and suppose also that the model $F$ is not exact. Then the output $y$ is surely infected by error, said the obtained output is $\tilde{y}=y+\delta y$.
My question is : How could one partition the output error into error coming from $F$ and  input error ? In other words, How could we write  the error  $\delta y$ as two parts, first part coming from the error of  $x$ and the second part coming from the error of  $F$?
Does anay one have an idea ? 
Any help would be highly appretiated. 
 A: A possible answer is that in fact the wiki article already sums it up pretty well: https://en.wikipedia.org/wiki/Propagation_of_uncertainty.
In general you need the partial derivatives, or in the multidimensional case the Jacobian (matrix). If you know the function, this is rather straightforward an exercise in calculus. In case of a linear function, the derivatives become trivial.
In the general case of an arbitrary function, you can also chose from several other options. The article lists five variants; my practical knowledge extends to two of these. The accuracy and feasibility depend on the type of function and the kind of approximation you need.
1) Series evolution: you can do the Taylor expansion and stop at your choice of order. This introduces a shift in the expectation value the propagated error w.r.t the true error (of an infinite series) -- it becomes biased. Given a well behaved function, this might be negligible.
2) Monte-Carlo simulation. Has the benefit that you might not even have to know the actual analytical (if existing) form of your function. But this becomes an own branch of research, depending on the function. Keywords here are efficiency and accuracy of the calculation.
The other methods are more specialized, mostly to problems that can't be well approximated by the two above.
So, concerning your original question: to disentangle the uncertainty $\delta y$, simply write down the full Taylor expansion term. For the first order (which is sufficient in most cases) have a look at the wiki page. 
