# How do I "propagate" the covariance?

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 $$X/Y$$: $$\mathrm{Cov}[X+Y,X/Y]$$?