Mathematically manipulating error resources

Hate asking such a general question, but I'm so unfamiliar with statistics in general that I don't know what to search or where to find the answer.

I'd like to learn how to manipulate errors. Like, if I had a function $f\left(x,y,z\right)$, and I knew $\sigma_x,\sigma_y,\sigma_z$, what the error of $\sigma_{f}$ is.

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Search for delta method –  whuber Mar 23 '12 at 4:17
Search for transforming random variables –  guest Mar 23 '12 at 4:23