Where does the error propagation formula comes from?

Question

As an engineering student I have come several times across the formula $$\sigma_{f(\vec{x})}=\sqrt{\sum_{i} \big (\dfrac{\partial f}{\partial x_{i}}\sigma_{x_{i}}\big )^{2}}$$ for the propagation of errors, but I never understood where does it come from. Could you provide some reference for me to understand it?

Answer 1

If you have just one error, $\dfrac{\partial f}{\partial x_{i}}\sigma_{x_{i}}$ gives the error in $f$. It essentially comes from the Taylor series, linearizing the value of $f$ around the correct value. If you have a number of errors, we think of them as uncorrelated, so it is a random walk. The root sum square is the expected distance of a random walk.