# Error propagation: when should I introduce self covariance?

I have a function $$z_{ik}$$ dependent from $$x_{ij}$$ with this form:

$$z_{ik}=(\sum_j x_{ij}\alpha_{jk}) \cdot ((\sum_jx^2_{ij})\beta_k)^{-1/2}$$ for $$i \in [1, N]$$, $$j \in [1, M]$$ and $$k \in [1, L]$$

The $$x_{ij}$$ are measured with an error $$\sigma_{ij}$$, while $$\alpha_{jk}$$ and $$\beta_k$$ are assumed to be without error.

I have to compute the error on $$z_{ik}$$, i.e. $$\sigma_{ik}$$. I have computed $$\frac{\partial z_{ik}}{\partial x_{ij}}$$, but I'm wondering if I should introduce the covariance between $$x_{in}$$ and $$z_{im}$$, for $$n, m \in [1, M]$$.