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I have two measurements, $X\pm\delta X$ and $Y\pm \delta Y$, and their "measured" covariance $\mathrm{Cov}[X,Y]$. (My case is that $\mathrm{Cov}[X,Y]=0$, but I want some general formulation if possible.)

If I were to find $F = f(x=X)$, then I would propagate like $\delta F \simeq \left.\frac{\partial f}{\partial x} \right|_{x=X}\delta X$.

My question is: Is it possible to do the same for covariance? If so, how?

For instance, how would I find the covariance between $X+Y$ and $X/Y$: $\mathrm{Cov}[X+Y,X/Y]$?

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