# analytical propagation of error

I have this function:

$$\rm \mu(R) = \mu_{\rm eff} + \frac{2.5 b_{\eta}}{ln(10)} \left[ \left( \frac{R}{R_{\rm eff}}\right)^{1/ \eta} - 1 \right]$$

where there are associated errors with $$\rm \mu(R)$$, (lets call it $$\rm \sigma_{\mu}$$), which depends on $$\rm R$$, $$\mu_{\rm eff}$$ ($$\rm \sigma_{\mu_{eff}}$$), and $$R_{\rm eff}$$ ($$\rm \sigma_{R_{eff}}$$). Additionally there is the parameter $$b_{\eta}$$ which does not have associated errors but is related with $$\eta$$ by $$b_{\eta} = 1.9992\eta - 0.3271$$.

The idea is to get the associated error of $$\eta$$. By solving for $$\eta$$ one gets the equation:

$$\rm \eta = \log(R/R_{eff}) \, / \, log[ln(10)(\mu(R) - \mu_{eff})/2.5b_{\eta} + 1]$$

Now, by assuming $$\rm \sigma_{b_{\eta}} = 0$$ and applying propagation of error from multivariable calculus one finds a very "annoying" equation (I wont post it due to its incredible big size). So, although I have the equation which will give me the uncertainty of $$\eta$$, I'm not quite sure if it is correct, because I'm not sure if I can just assume $$0$$ uncertainty for $$b_{\eta}$$ since its a function of $$\eta$$ itself.

Additionally I have another question. Imagining that the equation for the propagation of error is correct (so that I'm not doing a huge mistake by disregarding $$b_{\eta}$$), what would be the correct way to proceed from here since $$\rm \mu(R)$$ is a function of $$\rm R$$, so there is a different $$\rm \sigma_{\mu}$$ depending on $$\rm R$$? What I did was to assume for $$\rm R$$ and $$\rm \mu(R)$$ the point where $$\rm \sigma_{\mu}$$ reaches the maximum, having an estimate for the maximum error of $$\eta$$. Is this the way?