I got stuck with the derivative of the following function: $$\operatorname{erf}(\frac{\operatorname{logit}(\theta)-\mu}{\sqrt {2\sigma^2}})$$
with respect to $\theta$.
Are there handy approximations with elementary functions in that case?
Any help will be appreciated, thanks in advance!
The error function is defined by $\text{erf}(x)=\frac{1}{\sqrt{\pi}}\int_0^{x}e^{-t^2}\mathrm{d}t$. Therefore $\frac{\mathrm{d}}{\mathrm{d}x}\text{erf}(x)=\frac{2}{\sqrt{\pi}}e^{-x^2}$.
Set $x=\frac{\text{logit}(\theta)-\mu}{\sqrt{2\sigma^2}}$ and use the chain rule. Remember that $\text{logit}\theta$ is defined as $\frac{\theta}{1-\theta}$ which differentiates to $\frac{\mathrm{d}}{\mathrm{d}\theta}\text{logit}\theta=\frac{-1}{(1-\theta)^2}$.