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I got stuck with the derivative of the following function: $$erf(\frac{logit(\theta)-\mu}{\sqrt {2\sigma^2}})$$

with respect to $\theta$.

Are there handy approximations with elementary functions in that case?

Any help will be appreciated, thanks in advance!

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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 $\frac{\mathrm{d}}{\mathrm{d}\theta}\text{logit}\theta=\frac{-1}{(1-\theta)^2}$.

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  • $\begingroup$ Thanks a lot! The problem is that theta is in upper integration limit. Should I use Leibniz rule then? $\endgroup$ – Vladimir Feb 3 at 21:24

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