I'm given a quantity which is defined as $$a=\frac{bcd}{ef}$$ and I know the relative uncertainties in each of $b,c,d,e,f$. I'm supposed to find relative uncertainty in $a$.
As far as I have learnt, we have $$\%\frac{\Delta a}{a}=\%\frac{\Delta b}{b}+\%\frac{\Delta c}{c}+\%\frac{\Delta d}{d}+\%\frac{\Delta e}{e}+\%\frac{\Delta f}{f}$$ The $\%$ sign indicates that the relative uncertainties are in percentage.
However, is this actually correct, that is, are relative uncertainties really additive? I couldn't find a good reference which explicitly states so.
I think that's perhaps not the best way to handle the errors. For instance, if $e$ is small but has a large error this will greatly influence the error. Perhaps consider the linear approximation $L$ to the function? That's assuming that you can ignore second order errors.
We have: \begin{align*} a(b,c,d,e,f)&\approx L(b,c,d,e,f)\\ &=a(b_0,c_0,d_0,e_0,f_0)+\frac{\partial a}{\partial b}(b-b_0)+\frac{\partial a}{\partial c}(c-c_0)+\\ &+\frac{\partial a}{\partial d}(d-d_0)+\frac{\partial a}{\partial e}(b-b_0). \end{align*}
Therefore $$\left|\frac{\partial a}{\partial b}(b-b_0)+\frac{\partial a}{\partial c}(c-c_0)+\\ +\frac{\partial a}{\partial d}(d-d_0)+\frac{\partial a}{\partial e}(b-b_0)\right|$$ will give an approximation to the absolute uncertainty in $a$. Probably you'd estimate this using the triangle inequality.