# Where does the error propagation formula comes from?

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?

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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.
@Francisco: if you know all the errors will drive the value of $f$ in the same direction, you should add them linearly. This will give a larger (sometimes much larger) $\sigma_f$ It is the worst case, and in some applications it is insisted on. –  Ross Millikan Feb 16 '13 at 1:47