So I have been taught two formulas for error propagation:
For $Z=A+B$,
$\sigma_Z=\sqrt{(\sigma_A^2+\sigma_B^2)}$
and for Z=AB or Z=A/B
$(\dfrac{\sigma_Z}{Z})^2=(\dfrac{\sigma_A}{A})^2+(\dfrac{\sigma_B}{B})^2$
Are these not functions of 2 variables? Because I've also learnt the following:
$\sigma_Z=\dfrac{\partial Z}{\partial A}\sigma_A+\dfrac{\partial Z}{\partial B}\sigma_B$
So for example, which would I use for $Z=A-8B$?
One of the formulae in your question is not quite correct (as pointed out by Nikos M) but it should be
${\sigma_Z}^2 = ({\delta Z \over \delta A})^2 {\sigma_A}^2 + ({\delta Z \over \delta B})^2 {\sigma_B}^2$
This is the basic equation for propagating uncertainties.
The version you have in your question without the squared terms is a simplification of this, which strictly speaking is not correct, but can often be used and will be close to the correct answer.
The two formulae for the examples you give for $Z=A+B$ and $Z = AB$ or $A/B$ are correct and can be derived from the equation above.
In the case that $Z=kA$ where $k$ is a perfectly know constant $\sigma_Z = k \sigma_A$.
For $Z=A-B$ you use the same formulat as for $Z=A+B$ that you quote in your question.
Putting these two together gives for $Z=A-8B$
${\sigma_Z} = \sqrt{{\sigma_A}^2 + 8^2 {\sigma_B}^2}$
hence
${\sigma_Z} = \sqrt{{\sigma_A}^2 + 64 {\sigma_B}^2}$
It is worth noting that the fundamental euqation at the top needs to be used for cases which are more complicated - such as $Z=A/(A+B)$
References suggested by (and with many thanks to) Nikos M.
1 http://en.wikipedia.org/wiki/Variance
2 http://en.wikipedia.org/wiki/Taylor_expansions_for_the_moments_of_functions_of_random_variables
@Tom's answer appears absolutely correct, but omits to mention the assumptions behind these formulae:
(i) The measured values of $A$ and $B$ follow a normal distribution.
(ii) The measurement of $A$ does not depend on $B$ and vice-versa. i.e. the variables are independent.
(iii) Over the range $\pm \sigma_A$ and $\pm \sigma_B$; $Z(A)$ and $Z(B)$ are approximately linear functions.
For your example of $Z(A)=A$ and $Z(B)=-8B$ that is clearly the case, but the latter condition is very important and often forgotten. To see the truth of this work out the uncertainty in $Z = \sin(A) + \tan(B)$ when $A= B = (\pi/2\ \pm 0.1)$, where the approximate formula fails.
