How do I calculate the error propagation of this function? I'm using a molecular biological method which requires the following normalization
$$Q = \frac{A}{\sqrt[k]{\prod_{i=1}^k B_i}}.$$
How can I find the standard deviation of $Q$, given that $A$ and $B_i$ have known standard deviations? Unfortunately my math training is not good enough for me to solve this. Is there some kind of standard approach to solve this?
EDIT
The logarithm of the above expression is given by:
$$
\log(Q)=\log(A)-\frac{1}{k}\cdot \sum_{i=1}^{k}\log(B_{i})
$$
 A: If $(A, B_{1},\ldots, B_{k})$ are uncorrelated (i.e. $\mathrm{Cov}(A,B_{i})=0$ for all $i$ and  $\mathrm{Cov}(B_{i},B_{j})=0$ for all $i\neq j$), then the standard deviation of $\log(Q)$ is given by:
$$
\mathrm{SD}(\log(Q))=\sqrt{\mathrm{SD}(\log(A))^{2} + \left(\frac{1}{k}\right)^{2}\cdot \sum_{i=1}^{k}\mathrm{SD}(\log(B_{i}))^{2}}
$$
Derivation
Let's define $\mathrm{Var}(X)$ as the variance of $X$ and $\mathrm{SD}(X)$ as the standard deviation of $X$, where $\mathrm{SD}(X)=\sqrt{\mathrm{Var}(X)}$. Further, we can make use of the following properties of variances where $(X, Y, X_{1}, \ldots, X_{i})$ are uncorrelated variables and where $a$ denotes a constant:
$$
\begin{align}
\mathrm{Var}(aX) &= a^{2}\mathrm{Var}(X)\\
\mathrm{Var}(X+Y) &= \mathrm{Var}(X) + \mathrm{Var}(Y)\\
\mathrm{Var}(X-Y) &= \mathrm{Var}(X) + \mathrm{Var}(Y)\\
\mathrm{Var}(X-aY) &= \mathrm{Var}(X) + a^{2}\mathrm{Var}(Y)\\
\mathrm{Var}\left(\sum_{i}X_{i}\right) &= \sum_{i}\mathrm{Var}(X_{i})\\
\end{align}
$$
Our equation has the basic form $X-a\sum_{i}Y_{i}$, where $X=\log(A), a=\frac{1}{k}$ and $Y_{i}=\log(B_{i})$. Now we can put all pieces together:
$$
\mathrm{Var}(\log(Q))=\mathrm{Var}(\log(A)) + \left(\frac{1}{k}\right)^{2}\cdot \sum_{i=1}^{k}\mathrm{Var}(\log(B_{i}))
$$
And because the standard deviation is just the square root of the variance, we can substitute the variance by the squared standard deviation and take the sqare root on both sides and get the formula above.
